Motivic homotopy theory in derived algebraic geometry

Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) vorgelegt dem Fachbereich Mathematik der Universit¨atDuisburg-Essen

von Adeel Khan geboren in Kanada

Essen, July 2016

Contents

Preface 5 1. What is motivic homotopy theory? 6 2. Why derived schemes? 6 3. What we do in this text 7 4. Relation with previous work 7 5. What is not covered in this text 8 6. Acknowledgements 8

Chapter 0. Preliminaries 11 1. Introduction 12 2. (∞, 1)-categories 16 3. (∞, 2)-categories 26 4. Derived schemes 30 5. Local properties of morphisms 35 6. Global properties of morphisms 39 Chapter 1. Motivic spaces and spectra 43 1. Introduction 44 2. Motivic spaces 48 3. Pointed motivic spaces 53 4. Motivic spectra 55 5. Inverse and direct image functoriality 59 6. Smooth morphisms 61 7. Closed immersions 64 8. Thom spaces 72 9. The localization theorem 74 Chapter 2. The formalism of six operations 83 1. Introduction 84 2. Categories of coefficients 91 3. Motivic categories of coefficients 96 4. The formalism of six operations 111 5. Example: the stable motivic homotopy category 118 Bibliography 121

3

Preface

5 6 PREFACE

1. What is motivic homotopy theory?

1.1. The existence of a motivic cohomology theory was first conjectured by A. Beilinson [Bei87]. This cohomology theory was expected to be universal with respect to mixed Weil cohomologies like `-adic cohomology or algebraic de Rham cohomology; that is, there should be cycle class maps from the rational motivic cohomology groups to, say, `-adic cohomology. Further, this cohomology theory was expected to arise from a category of coefficients, the category of so-called motivic complexes. Hence for each scheme S, Beilinson conjectured the existence of categories DM(S), admitting Grothendieck’s formalism of six operations: there ∗ ! should be operations (f , f∗, f!, f , ⊗, Hom) acting on these categories, for f :T → S a morphism of schemes, satisfying various compatibilities like base change and projection formulas. After the work of many mathematicians including J. Ayoub, D.-C. Cisinski, F. D´eglise,M. Levine, and V. Voevodsky, we are close to the theory envisioned by Beilinson: we have categories of motivic complexes over general base schemes, satisfying the formalism of six operations at least partially. We refer the reader to [CD09] for the state of the art.

1.2. In algebraic topology, the analogue of motivic complexes are complexes of abelian groups: the derived category D(Ab) is the category of coefficients for singular cohomology. It turns out that D(Ab) is the linear version of another category Spt (whose objects are called spectra), in the sense that it can be identified with the category of modules over a spectrum ZSpt in Spt. In the category Spt, we have representability not only of singular cohomology, but of generalized cohomology theories like complex K-theory and complex cobordism. We can form an analogue of the category Spt in algebraic geometry. F. Morel, and Voevodsky constructed for each scheme S, categories SH(S) of motivic spectra. The thesis of Ayoub [Ayo07] established the construction of the six operations on motivic spectra, following ideas of Voevodsky.

Just as in topology, the category of modules over the spectrum ZS, representing motivic cohomology, is equivalent to the category DM(S) (at least when S is the spectrum of a field of characteristic zero; see [RØ08]). In the category SH(S) we have representability of generalized motivic cohomology theories like homotopy invariant algebraic K-theory and algebraic cobordism. Mixed Weil cohomologies are also representable by motivic spectra (see [CD12]). When a cohomology theory is representable as a motivic spectrum, we may view cohomology classes as morphisms in the category SH(S). This gives us the possibility to take advantage of the power of the formalism of six operations, of which only shadows can be seen at the level of cohomology groups (K¨unnethformulas, Gysin maps, etc.).

2. Why derived schemes?

2.1. As was expected by Beilinson, motivic cohomology is closely related to intersection theory and algebraic cycles. For example, Levine proved that over a field, it can be computed as the higher Chow groups defined by S. Bloch (see [Lev06]). It is not currently known how to extend this comparison to more general bases, as conjectured by Beilinson; Levine’s proof uses a highly technical moving lemma which is only known for fields. This is related to the lack of a good intersection theory over general base schemes (say regular and of finite type over Spec(Z)), a problem posed by A. Grothendieck in [SGA 6] and studied indirectly by D. Quillen and R. Thomason in their work on algebraic K-theory. We will not have anything to say about this problem in this thesis, but it was the motivation for us to pass to the world of derived algebraic geometry. 4. RELATION WITH PREVIOUS WORK 7

2.2. Derived algebraic geometry is an extension of classical algebraic geometry where we allow schemes to have “higher nilpotents”. For example, let Z and Z0 be closed subschemes of a regular scheme X, defined by ideals I and I0, respectively. Let us assume that Z and Z0 intersect properly, so that codim(Z ∩ Z0, X) = 0 0 codim(Z, X) + codim(Z , X). It is well-known that the scheme-theoretic intersection Z ×X Z does not capture the correct intersection multiplicity in the non-transverse case. Rather, according to Serre’s intersection formula (see [Ser00]), we must take all the groups

OX,x 0 Tori (OX,x/Ix, OX,x/Ix)(i > 0) 0 into account, for each generic point x of Z ×X Z . On the other hand, we can take the intersection in the world of derived algebraic geometry, R 0 0 i.e. the derived fibred product Z ×X Z . Its underlying classical scheme coincides with Z ×X Z , but when Z and Z0 do not intersect transversally, it has higher nilpotents. More precisely,

0 its structure sheaf is a sheaf of simplicial commutative rings; we recover the sheaf OZ ×X Z as π0(O R 0 ), but it also has higher homotopy groups Z ×X Z

πi(O R 0 )(i 0) Z ×X Z > whose stalks are none other than the Tor groups written above. The data contained in these higher homotopy groups is what we think of as “higher” nilpotents. This observation is what suggested to us that derived algebraic geometry could be a more natural setting for the study of motives.

3. What we do in this text

In this thesis we will show that the motivic homotopy categories SH(S) extend to the world of derived algebraic geometry, as well as the full formalism of six operations. In Chapter 0, we review some preliminaries about theories of (∞, 1)-categories and derived algebraic geometry. The reader who is already familiar with these theories may want to skim this chapter in order to acquaint himself with the notation we use. In Chapter 1, we construct the unstable and stable motivic homotopy categories over a derived scheme. We prove our first main result, the analogue of Morel–Voevodsky’s localization theorem for motivic spaces. In Chapter 2, we construct the formalism of six operations on the stable motivic homotopy category. We use the formalism of (∞, 2)-categories of correspondences introduced in [GR16].

4. Relation with previous work

4.1. Motivic homotopy theory over classical base schemes was introduced in [MV99], using the language of model categories. An (∞, 1)-categorical construction was given in [Rob14]. Our definition is essentially a straightforward generalization to the setting of derived schemes. The only subtlety is that, unlike Morel–Voevodsky, we do not impose any finiteness conditions on our base schemes. This means that descent (by which we mean descent with respect to Cechˇ covers) is no longer equivalent to hyperdescent (by which we mean descent with respect to arbitrary hypercovers). Here we work with the weaker notion of descent. The main result of Chapter 1, the localization theorem, was proved by Morel–Voevodsky in loc. cit. Our proof follows the same general strategy, but since we impose a weaker descent condition on our sheaves, we have to work a bit more. 8 PREFACE

4.2. The formalism of six operations in motivic homotopy theory, in the setting of classical schemes, has been constructed by J. Ayoub in his thesis [Ayo07], following ideas of Voevodsky. We use part of his work as input in our construction. However, Ayoub works with the language of triangulated categories, and so does not obtain the full homotopy-coherent system of compatibilities between the various operations. An (∞, 1)-categorical lift of the six functor formalism was completed in the thesis of Robalo [Rob14], using the framework of [LZ12] based on multi-simplicial sets. In this work, we follow a different approach, developed by Gaitsgory–Rozenblyum in [GR16], based on (∞, 2)-categories of correspondences. The relation between the two is explained in loc. cit. However, we slightly modify the approach of Gaitsgory–Rozenblyum in order to encode projection formulas; their method of encoding projection formulas cannot be applied to the category of motivic spectra, because we do not know if the canonical functors SH(X) ⊗SH(S) SH(Y) → SH(X ×S Y) are equivalences.

5. What is not covered in this text

5.1. Using the framework we set up in this text, it is possible to give a definition of motivic cohomology, homotopy invariant algebraic K-theory, and algebraic cobordism of derived schemes. One can also prove that these cohomology theories are insensitive to higher nilpotent thickenings (just as they are known to be insensitive to usual nilpotent thickenings). In a sequel to this work we will apply this to construct virtual fundamental classes in oriented generalized motivic Borel–Moore homology theories.

5.2. In this text we work with the version of derived algebraic geometry where commutative rings are replaced by simplicial commutative rings. It is possible to consider other contexts for derived algebraic geometry, like spectral algebraic geometry, where one uses connective E∞-ring spectra. Another possibility, lying between simplicial commutative rings and connective E∞-ring spectra, is connective E∞-dg-algebras over a commutative ring. The results of Chapter 1 are true in all of these contexts, and the proofs we have provided generalize mutatis mutandis; see [Kha16]. There is some subtlety however: there are two possible notions of smoothness in spectral algebraic geometry, corresponding to two possible choices for the affine line. Spectral smoothness, the version of smoothness based on the cotangent complex, corresponds to the spectral affine line. On the other hand, classical smoothness, by which we mean flat and smooth on underlying classical schemes, corresponds to the classical affine line. The correct definition of the motivic homotopy category turns out to be the one using the spectrally smooth site and the spectral affine line. However, a surprising observation is that, over a classical scheme, the motivic homotopy category may not coincide with the Morel–Voevodsky motivic homotopy category; for example, over any field of positive characteristic, the spectrally smooth site is different from the usual smooth site.

6. Acknowledgements

I would like to thank the many people who were directly or indirectly involved with this project. I am immensely grateful to my advisor Marc Levine for his support and interest in this project. I am equally grateful to Denis-Charles Cisinski, who originally suggested to me that the localization theorem should hold in this setting. I thank him for the countless ideas he shared with me and for his patience and generosity in answering my questions. 6. ACKNOWLEDGEMENTS 9

I would like to thank Marc Hoyois for his patience in answering questions about (∞, 1)- categories and motivic homotopy theory. I would like to thank Bertrand To¨enfor his patience in answering questions about derived algebraic geometry. I would like to thank Timo Sch¨urgfor first introducing me to model categories and derived algebraic geometry. I would like to thank the following mathematicians for useful and interesting discussions about this work: Ben Antieau, Fr´ed´ericD´eglise,Brad Drew, Elden Elmanto, Lie Fu, Dennis Gaitsgory, David Gepner, Paul Arne Østvaer, Fr´ed´ericPaugam, Mauro Porta, Marco Robalo, and Markus Spitzweck. I would like to thank the following mathematicians for generally useful and interesting discussions about mathematics: Peter Arndt, Pieter Belmans, John Calabrese, Andrea Gagna, Edoardo Lanari, Zhen Lin Low, Sean Tilson, and Niels uit de Bos. I am particularly grateful to Elden Elmanto and Simon Pepin Lehalleur for their careful reading of drafts of this work. I am grateful to those who made the department a pleasant place to work, including Andrea, Chiara, Federico, Gabriela, Ishai, Jin, Lorenzo, Mura, Niels, Toan, and Vova. Thanks also to Frau Meinel for her help with bureaucracy. Most of all, I thank my family for their boundless love and support.

CHAPTER 0

Preliminaries

1. Introduction 12 1.1. (∞, 1)-categories 12 1.2. Derived algebraic geometry 13 1.3. Conventions 14 1.4. Organization of this chapter 14 2. (∞, 1)-categories 16 2.1. (∞, 1)-categories 16 2.2. Monomorphisms 17 2.3. Presheaves 18 2.4. Filtered colimits 19 2.5. Sifted colimits 20 2.6. Arenas 21 2.7. Module arenas 22 2.8. Toposes 23 2.9. Sites 24 2.10. Stable categories 25 3. (∞, 2)-categories 26 3.1. (∞, 2)-categories as complete Segal spaces 26 3.2. Passage to right/left adjoints 26 3.3. Adjointable squares 27 3.4. The 2-category of arena modules 28 4. Derived schemes 30 4.1. Simplicial commutative algebra 30 4.2. Prestacks 31 4.3. Quasi-coherent sheaves 31 4.4. Stacks 32 4.5. Schemes 33 4.6. Closed immersions 33 4.7. Vector bundles 34 5. Local properties of morphisms 35 5.1. Square-zero extensions 35 5.2. The cotangent sheaf 36 5.3. Smooth and ´etalemorphisms 37 5.4. Deformation along square-zero extensions 38 5.5. Cobase change along nil-immersions 38 5.6. Lifting smooth morphisms along closed immersions 39 6. Global properties of morphisms 39 6.1. Proper morphisms 39 6.2. Closure 40 6.3. Compactifications of morphisms 41

11 12 0. PRELIMINARIES

1. Introduction

In this chapter we give a very brief survey of the theories of (∞, 1)-categories and derived algebraic geometry.

1.1. (∞, 1)-categories. Let R be a commutative ring. Recall that for R-modules M and N, R L the groups Tori (M, N) are shadows of a chain complex M ⊗R N which is well-defined only up to quasi-isomorphism: it depends on a choice of projective resolution. In other words, the abelian category of chain complexes of R-modules is too rigid for the purposes of homological algebra; categorical constructions like limits and colimits will not behave well with respect to quasi-isomorphism. Instead, the language of homotopy theory is appropriate. An abstract homotopy theory is the datum of a category C together with a class W of morphisms called weak equivalences. Such a pair (C, W) is a presentation of an (∞, 1)-category. For example, the (∞, 1)-category of homotopy types or spaces is presented equivalently by topological spaces (up to weak homotopy equivalence) or simplicial sets (also up to weak homotopy equivalence), among many other possibilities. In fact, the collection of homotopy theories is itself a homotopy theory, and this homotopy theory is equivalent to the homotopy theory of (∞, 1)-categories1. In particular, the homotopy theory of homotopy theories can be taken as a model for (∞, 1)-categories. However, this approach can be problematic. After all, a pair (C, W) is itself only well-defined up to weak equivalence of homotopy theories2, which is much weaker than equivalence of the underlying category C, and it is not clear how to extract (∞, 1)-categorical information like (co)limits from it, as they cannot just be computed as (co)limits in the category C. Even more simply, morphisms in the associated (∞, 1)-category are not just morphisms in the category C. Classically, these problems have been solved by using D. Quillen’s theory of model categories. A model structure on a pair (C, W) is the data of classes of cofibrations and fibrations subject to various axioms. The purpose of this structure is to allow computation of (∞, 1)-categorical (co)limits as usual categorical (co)limits in the category C, after suitable (co)fibrant replacement of the diagram. For example, there are projective and injective model structures on the category of chain complexes, where cofibrant or fibrant replacement corresponds to choosing projective or injective resolutions, respectively. Similarly, sets of morphisms in the (∞, 1)-category can be computed as the set of morphisms in the category C, after taking cofibrant and fibrant replacements of the objects. Though this approach is very powerful, the choice of a specific model-categorical presentation adds a factor of arbitrariness to all constructions and proofs, and requires some additional effort in checking that all constructions performed are in fact the homotopically correct ones. This effort can be nontrivial.3 The modern approach consists in forgetting about presentations entirely, and instead choosing a more transparent model for (∞, 1)-categories, like the quasi-categories of A. Joyal or complete Segal spaces of C. Rezk. For example, a quasi-category is by definition a simplicial set satisfying the weak Kan condition; an object of a quasi-category is a 0-simplex, a morphism is a

1This can be made into a mathematically rigourous statement after the work of [BK12]. 2Weak equivalence of homotopy theories is precisely equivalence of (∞, 1)-categories. 3The theory gets very technical very quickly. For example, let C be a small (∞, 1)-category. The (∞, 1)- category of presheaves on C can be presented by the injective model structure on the category of simplicially enriched functors between the simplicially enriched category associated to Cop, and the simplicially enriched category of simplicial sets. Compare with Footnote 4. 1. INTRODUCTION 13

1-simplex, and a functor between quasi-categories is a morphism of simpicial sets.4 Similarly, (co)limits and other categorical constructions have intrinsic quasi-categorical descriptions. Let us consider a simple example. Suppose for instance that we have a diagram

X0

X1 X in the (∞, 1)-category of spaces. The classical way to compute its limit would be choose a presentation of the (∞, 1)-category of spaces, for example by the model category of topological spaces, and to compute the homotopy limit in this model structure. Hence one would first replace this diagram by a diagram in the category of topological spaces, where each of the objects is fibrant (in the chosen model structure), and at least one of the morphisms Xi → X is a fibration (in the chosen model structure). A topological space with the homotopy type of the homotopy limit would then be given by the usual categorical limit of the resulting diagram. In the (∞, 1)-categorical approach, the limit of the diagram is simply the space of pairs (x0, x1, α), where xi is a point of Xi and α is a path in the space X between the images of the points x0 and x1. More precisely, each xi is a morphism pt → Xi, where pt denotes the terminal object in the (∞, 1)-category of spaces, and α is a commutative square

x0 pt X0

x1

X1 X. This is nothing more than the universal property of the (∞, 1)-categorical limit.

1.2. Derived algebraic geometry. The (∞, 1)-category of simplicial commutative rings is an enlargement of the ordinary category of commutative rings where we have objects like derived L tensor products A ⊗R B, for a commutative ring R and R-algebras A and B. By carefully replacing commutative rings by simplicial commutative rings in the definition of scheme (and hence passing to the world of (∞, 1)-categories), we obtain the definition of derived scheme. Any derived scheme S has an underlying classical scheme Scl, and the relationship between S and Scl is analogous to the relationship between the classical scheme Scl and its underlying reduced scheme Scl,red. In the same way that natural constructions involving algebraic varieties force one into the world of schemes, derived schemes also arise naturally from considerations involving their classical counterparts. For example, let S = Spec(R) be an affine scheme and let X = Spec(A) and Y = Spec(B) be affine schemes over S. The derived fibred product R L X × Y = Spec(A ⊗ B) S R is an important example of a derived scheme. When X or Y is flat over S, this coincides with the classical fibred product X ×S Y = Spec(A ⊗R B). In general, the derived tensor product contains information that cannot be recovered from the ordinary tensor product (cf. Serre’s intersection formula, which in certain situations computes intersection multiplicities in terms of R the groups Tori (A, B)).

4 op The quasi-category of presheaves on a small quasi-category C is the internal hom HomsSets(C , Spc), where Spc is the quasi-category of spaces. Compare with Footnote 3. 14 0. PRELIMINARIES

5 For another example , let S = Spec(R) be an affine scheme and f0, . . . , fn ∈ R a set of global sections. The closed derived subscheme cut out by the equations fi is defined as L Z = Spec(R /(fi)i), L where we have written R/ (fi)i for the Koszul complex of the sequence (fi)i, which can be defined as a simplicial commutative ring. When (fi)i is a regular sequence, the Koszul complex is quasi-isomorphic to the usual quotient R/(fi)i, so Z is identical to the classical scheme Spec(R/(fi)i), which is in this case a regularly embedded or lci closed subscheme of S. In general, Z is a derived scheme with underlying classical scheme

Zcl = Spec(R/(fi)i).

These examples can both be globalized to non-affine base schemes S. To some extent, they explain the ubiquity of flatness and regularity/lci assumptions in algebraic geometry: these are precisely the conditions that guarantee that the relevant derived schemes are actually classical.

1.3. Conventions. We will use the language of (∞, 1)-categories freely throughout the text. Though we will use the language in a model-independent way, we fix for concreteness the model of quasi-categories as developed by A. Joyal and J. Lurie. For simplicity we adopt the following conventions: The term “category” means, by default, “(∞, 1)-category” (= quasi-category). When we want to refer to an ordinary category, we will use the term “ordinary category” or “(1, 1)- category”. We will say that a morphism in an (∞, 1)-category is invertible or an isomorphism (as in [Joy04] and [GR16]) where some authors might use the word equivalence (e.g. [Lur09b]). We will use the symbol “=” for isomorphic objects in an (∞, 1)-category, as the notion of equality simply does not exist. The term “2-category” means, by default, “(∞, 2)-category”. When we want to refer to an ordinary 2-category, we will use the term “(2, 2)-category” or “ordinary 2-category”. For us, this term always means weak 2-category (a.k.a. bicategory). An ordinary 2-category in which the 2-morphisms are invertible will be called a “(2, 1)-category”. The term “scheme” means, by default, “derived scheme”. When we want to refer to a classical scheme, we will use the term “classical scheme”. Our focus is this chapter is on giving statements. We only attempt to give rigourous proofs when we do not know a reference in the literature. In particular, in this chapter, we will “define” (∞, 1)-categories and (∞, 1)-functors only on objects and 1-morphisms; the reader will find their precise constructions in the literature.

1.4. Organization of this chapter. This chapter is a very brief survey of some parts of the theories of (∞, 1)-categories and of derived algebraic geometry that we will use in this text. In Sect. 2 we briefly review several notions from the theory of (∞, 1)-categories that will play an important role in this text: stable (∞, 1)-categories, presheaves, filtered and sifted colimits, arenas (more commonly known as presentable (∞, 1)-categories), toposes, and sites. References for this section are [Lur09b], [Lur16], and [Joy04]. In Sect. 3 we review some (∞, 2)-category theory, and define an (∞, 2)-category Arenamod which will play an important role in Chapter 2. We follow the treatment of [GR16].

5The reader will observe that this is in fact a special case of the first example. 1. INTRODUCTION 15

In Sects. 4–6 we give a very brief exposition of the theory of derived schemes. We follow the functorial approach of To¨en–Vezzosi, see [TV08] and [MT10]. 16 0. PRELIMINARIES

2. (∞, 1)-categories

References for this section are [Lur09b], [Lur16], and [Joy04].

2.1. (∞, 1)-categories.

2.1.1. In the world of (∞, 1)-categories, all sets are replaced by spaces. Hence instead of elements of sets, we have points in spaces; instead of equality between two elements, we have paths between points, and so on. For example, in any (∞, 1)-category C, we have for any two objects x and y a mapping space

MapsC(x, y) ∈ Spc in the (∞, 1)-category of spaces.

2.1.2. Any ordinary category can be viewed as an (∞, 1)-category, with the same objects and discrete mapping spaces. This defines a fully faithful functor from the (∞, 1)-category of ordinary categories to the (∞, 1)-category of (∞, 1)-categories. Conversely, given an (∞, 1)-category C, we can form its underlying ordinary category, denoted (C)ordn, with the same objects and with morphisms given by connected components of mapping spaces:

Hom(C)ordn (x, y) = π0(MapsC(x, y)) (x, y ∈ C). The assignment C 7→ (C)ordn defines a left adjoint to the above inclusion.

2.1.3. In the sequel, the term “category” will mean “(∞, 1)-category” by default; when we need to refer to the classical notion of category, we will say “ordinary category”.

2.1.4. We now recall some standard features of ordinary category theory which have analogues in this world.

2.1.5. Given a category C, there are objects and morphisms. Morphisms can be invertible, or isomorphisms. There is a category Arrows(C) whose objects are morphisms in C, and morphisms are commutative squares.

2.1.6. A category C has an underlying space, denoted (C)Spc. The category of spaces embeds fully faithfully into the category of categories, and the assignment C 7→ (C)Spc is right adjoint to the inclusion. We also use the term ∞-groupoid for a category in the essential image of this embedding.

2.1.7. Given a category C, there is an opposite category (C)op obtained by reversing the directions of morphisms.

2.1.8. Given two categories C and D, we have a category Funct(C, D) whose objects are functors u : C → D. There are notions of fully faithful, essentially surjective, or equivalence, for any such functor. A functor is exact (resp. right-exact, left-exact) if it commutes with finite colimits and limits (resp. with finite colimits, resp. with finite limits).

2.1.9. Given a category C, there is a notion of full subcategory of C. 2. (∞, 1)-categories 17

2.1.10. Given two categories C and D, there is a notion of adjunction (u : C → D, v : D → C). Given an adjunction as above and a morphism f : u(c) → d with c an object of C and d an object of D, we say that the corresponding morphism g : c → v(d) is obtained by right transposition from f. Conversely, f is obtained by left transposition from g.

2.1.11. Given a category C and an object c, there are slice categories C/c and c\C. More generally, given a functor u : I → C, there are slice categories C/u and u\C.

Also, given functors u : C1 → C and v : C2 → C, there is a comma category (u ↓ v) whose objects are triples (c1, c2, f), with f a morphism u(c1) → v(c2) in C.

Taking C1 (resp. C2) to be the trivial category, the functor u (resp. v) defines an object c ∈ C, and we write the comma category (u ↓ v) as (c\v) (resp. (u/c)). Note that when we further take v (resp. u) to be the identity functor of C, we recover the slice categories (c\idC) = c\C (resp. (idC/c) = C/c).

2.1.12. Let I be a category. Given an I-indexed diagram (ci)i∈I in a category C, i.e. a functor (i ∈ I) 7→ (c ∈ C), there is a notion of colimit (resp. limit), denoted lim (resp. lim ), i −→i∈I ←−i∈I satisfying the expected universal property.

2.1.13. The collection of (small) categories forms a (large) category (∞, 1)-Cat.

2.1.14. Given a category C, there is a notion of simplicial object in C, which is a functor op (∆) → C. Here ∆ is the ordinary category whose objects are sets [n] := {0, 1 . . . , n} (n > 0), and morphisms are order-preserving morphisms of sets.

2.1.15. There is a notion of (closed) symmetric monoidal structure on a category C. Any category admitting finite products admits a canonical symmetric monoidal structure; when equipped with this structure, we call such a category a cartesian monoidal category. In a symmetric monoidal category, there is a notion of commutative monoid. We write CMon(C) for the category of commutative monoids. Given a commutative monoid O in C, there is a notion of O-module object in C. We write O-mod for the category O-modules.

2.1.16. A category is contractible if its underlying space is (weakly) contractible.

2.2. Monomorphisms.

2.2.1. Let X be a space. We say that X is n-truncated (n > 0) if πi(X, x) = 0 for i > n and any base point x. By convention, X is (−1)-truncated if it is empty or contractible, and (−2)-truncated if if is contractible. A morphism of spaces f :X → Y is n-truncated if its fibres are n-truncated.

2.2.2. In a category C, we say that an object c is n-truncated if for every object d in C, the mapping space MapsC(d, c) is n-truncated. A morphism f : x → y is n-truncated if for every object c in C, the morphism of spaces MapsC(c, x) → MapsC(c, y) is n-truncated.

2.2.3. We say that a morphism f in a category C is a monomorphism if it is (−1)-truncated. Note that a morphism is (−2)-truncated if and only if it is an isomorphism. 18 0. PRELIMINARIES

2.2.4. If C admits finite limits, then a morphism f : x → y is n-truncated (n > −1) if and only if the diagonal morphism x → x ×y x is (n − 1)-truncated.

2.2.5. The full subcategory of n-truncated objects (n > −2) in a category C is stable under small limits.

2.3. Presheaves.

2.3.1. Let C be a small category. A presheaf (of spaces) on C is a functor (C)op → Spc. We let P(C) denote the category of presheaves on C, which is by definition the functor category Funct((C)op, Spc).

2.3.2. Any object c ∈ C represents a presheaf hC(c) defined by

hC(c)(d) := MapsC(d, c) for each object d ∈ C.

The assignment c 7→ hC(c) defines a canonical functor hC : C → P(C), called the Yoneda embedding. The Yoneda lemma states: Proposition 2.3.3. Let F be a presheaf on C. For every object c ∈ C, there is a canonical isomorphism of spaces

MapsP(C)(hC(c), F) → F(c).

In particular, the Yoneda embedding hC is fully faithful. We say that a presheaf F is representable if it is isomorphic to hC(c) for some object c ∈ C.

2.3.4. The category P(C) admits arbitrary small colimits and limits, and they can be computed objectwise. That is, for any diagram of presheaves i 7→ Fi, indexed on a category I, we have canonical isomorphisms of spaces (lim F )(c) = lim F (c), −→ i −→ i i∈I i∈I (lim F )(c) = lim F (c), ←− i ←− i i∈I i∈I for each object c ∈ C.

2.3.5. In particular, the category P(C) admits a cartesian monoidal structure, i.e. a symmetric monoidal structure where the monoidal product is given by the cartesian product.

2.3.6. The category of presheaves admits the following universal property.

Given two categories C and D where small colimits are representable, we write Funct !(C, D) for the full subcategory of Funct(C, D) spanned by functors that commute with small colimits. Then we have: Proposition 2.3.7. Let C be a small category. For any category D where small colimits are representable, the canonical functor

(2.1) Funct !(P(C), D) → Funct(C, D), given by restriction along the Yoneda embedding, is an equivalence. 2. (∞, 1)-categories 19

In other words, the category P(C) is freely generated by C under small colimits.

In particular, any functor u : C → D admits a canonical extension to a functor u! : P(C) → D. This is left adjoint to the functor u∗ of restriction of presheaves along u.

We call u! the left Kan extension of u. For any presheaf F, we have a canonical isomorphism (2.2) u (F) = lim u(c), ! −→ hC(c)→F where the colimit is taken over the slice category (hC/F).

2.3.8. Taking D = P(C) above, we see that the identity functor of P(C) is the left Kan extension of the Yoneda embedding. In particular, every presheaf F can be canonically identified with a small colimit of repre- sentable presheaves: (2.3) F = lim h(c). −→ hC(c)→F

2.3.9. Let u : C → D be a functor of small categories. Then the universal property of the category of presheaves gives a canonical functor

u! : P(C) → P(D), left adjoint to the restriction functor u∗ :F 7→ F ◦ u.

The functor u! is the unique functor which fits in the commutative square C u D

hC hD P(C) u! P(D) and commutes with small colimits.

2.4. Filtered colimits.

2.4.1. Let I be a small category. Definition 2.4.2. The category I is κ-filtered, for a regular cardinal κ, if I-indexed colimits commute with κ-small limits, in the category of spaces.

The category I is filtered if it is ℵ0-filtered, i.e. I-indexed colimits commute with finite limits.

Any ordinary category is filtered in the sense of [SGA 4] if and only if it is filtered when viewed as an (∞, 1)-category. Every filtered category is contractible.

2.4.3. Let C be a small category. We define:

Definition 2.4.4. A presheaf F on C is κ-inductive if the slice category (hC/F) is κ-filtered.

The formula (2.3) shows that a presheaf is κ-inductive if and only if it is a κ-filtered colimit of a diagram of representable presheaves. Note that representable presheaves are κ-inductive.

In the case κ = ℵ0, we say simply inductive instead of ℵ0-inductive. 20 0. PRELIMINARIES

2.4.5. Let Ind κ(C) (resp. Ind(C)) for the full subcategory of P(C) spanned by κ-inductive presheaves (resp. by inductive presheaves). 0 0 Given two categories D and D admitting small κ-filtered colimits, we write Funct κ-filt(D, D ) ⊂ Funct(D, D0) for the full subcategory spanned by functors commuting with κ-filtered colimits.

We have the following universal property for Ind κ(C): Proposition 2.4.6. Let C be a small category and D a category admitting small κ-filtered colimits. The canonical functor

Funct κ-filt(Ind κ(C), D) → Funct(C, D), given by restriction along the Yoneda embedding, is an equivalence.

In particular any functor u : C → D extends uniquely to a functor u! : Ind κ(C) → D. This is also given by the formula (2.2), and we also call it the left Kan extension of u.

2.5. Sifted colimits.

2.5.1. Let I be a small category. Definition 2.5.2. (i) The category I is κ-sifted, for a regular cardinal κ, if I-indexed colimits commute with κ-products, in the category of spaces.

(ii) The category I is sifted if it is ℵ0-sifted, i.e. I-indexed colimits commute with finite products.

Any κ-filtered category is κ-sifted. The category ∆op is sifted. Any ordinary category is sifted in the sense of [GU71] if and only if it is sifted when viewed as an (∞, 1)-category. Every sifted category is contractible.

2.5.3. Let C be a small category. We define:

Definition 2.5.4. A presheaf F on C is weakly κ-inductive if the slice category (hC/F) is κ-sifted.

The formula (2.3) shows that a presheaf is weakly κ-inductive if and only if it is a κ-sifted colimit of a diagram of representable presheaves. If C admits κ-small coproducts, a presheaf is weakly κ-inductive if and only if it commutes with κ-small products. Note that representable presheaves are weakly κ-inductive.

In the case κ = ℵ0, we say simply weakly inductive instead of weakly ℵ0-inductive.

2.5.5. Let Wind κ(C) (resp. Wind(C)) for the full subcategory of P(C) spanned by weakly κ-inductive presheaves (resp. by weakly inductive presheaves). 0 0 Given two categories D and D admitting small κ-sifted colimits, we write Funct κ-sift(D, D ) ⊂ Funct(D, D0) for the full subcategory spanned by functors commuting with κ-sifted colimits.

We have the following universal property for Wind κ(C): Proposition 2.5.6. Let C be a small category and D a category admitting small κ-sifted colimits. The canonical functor

Funct κ-sift(Wind κ(C), D) → Funct(C, D), given by restriction along the Yoneda embedding, is an equivalence. 2. (∞, 1)-categories 21

In particular any functor u : C → D extends uniquely to a functor u! : Wind κ(C) → D. This is also given by the formula (2.2), and we also call it the left Kan extension of u.

2.6. Arenas. The terminology “arena” was introduced by [Joy04]. In [Lur09b] the term “presentable ∞-category” is used instead.

2.6.1. Let C and D be categories where small colimits are representable, and κ a regular cardinal. A functor C → D is κ-accessible if it commutes with κ-filtered colimits. It is accessible if it is κ-accessible for some κ. Definition 2.6.2. A left localization of a category C is a functor γ : C → D admitting a fully faithful right adjoint. An accessible localization (resp. κ-accessible localization) is a left localization such that the right adjoint of γ is accessible (resp. κ-accessible). An exact localization is an accessible localization such that γ is exact (i.e. commutes with finite limits).

2.6.3. Let C be a category. We define:

Definition 2.6.4. The category C is compactly generated if there exists a small category C0 and an ℵ0-accessible localization γ : P(C0) → C.

For example, for any small category C0, the category Ind(C0) is compactly generated.

2.6.5. Let C be a category. The following notion will play an important role throughout this text:

6 Definition 2.6.6. The category C is an arena if there exists a small category C0 and an accessible localization γ : P(C0) → C. A morphism of arenas is a functor that commutes with small colimits.

We write Arena for the category of arenas.

2.6.7. The class of arenas is stable under the formation of slice categories and functor categories: Lemma 2.6.8. Let C be an arena. For any functor p : I → C, the slice categories (C/p) and (p\C) are arenas.

Lemma 2.6.9. Let C and D be arenas. Then the category Funct !(C, D) of morphisms of arenas is an arena.

2.6.10. The following fact will be referred to as the “adjoint functor theorem”: Proposition 2.6.11. Let C and D be arenas. Then a functor u : C → D admits a right adjoint if and only if it commutes with small colimits. It admits a left adjoint if and only if commutes with small limits and is accessible.

We also have: Proposition 2.6.12. Let C be an arena. Then a functor (C)op → Spc is representable if and only if it commutes with small limits.

6This term is due to Joyal [Joy04]. The term presentable or locally presentable (∞, 1)-category is more common in the literature. 22 0. PRELIMINARIES

2.6.13. Let S be a set of morphisms in an arena C. We define:

Definition 2.6.14. (i) An object c ∈ C is S-local if, for every morphism f : x → y in S, the induced morphism of spaces

MapsC(y, c) → MapsC(x, c) is invertible. (ii) A morphism f : x → y is an S-local equivalence if for every S-local object c, the induced morphism of spaces

MapsC(y, c) → MapsC(x, c) is invertible.

We have:

Proposition 2.6.15. Let C be an arena. For any essentially small7 set S of morphisms in C, the inclusion of the full subcategory CS of S-local objects admits a left adjoint L: C → CS, which exhibits CS as an accessible left localization of C.

In the above situation, a morphism f in C induces an isomorphism L(f) if and only if f is an S-local equivalence. In fact, all accessible left localizations of an arena C arise in the above way.

2.6.16. Let C be an arena and S an essentially small set of morphisms.

Given an arena D, let Funct !,S(C, D) denote the full subcategory of Funct !(C, D) spanned by functors that send morphisms in S to isomorphisms in D.

We have the following universal property of CS:

Proposition 2.6.17. For any arena D, the canonical morphism

∼ Funct !(CS, D) −→Funct !,S(C, D) given by restriction along the functor L: C → CS, is an equivalence.

2.6.18. Another way to formulate the above is as follows. Consider the category of pairs (C, S), with C an arena and S an essentially small set of morphisms. There is a canonical fully faithful functor D 7→ (D, iso), where iso is the set of isomorphisms in D. It admits a right adjoint, the forgetful functor (C, S) 7→ C.

It also admits a left adjoint, the left localization functor (C, S) 7→ CS. Further, both adjoints are symmetric monoidal, with respect to the cartesian monoidal structure on the category of pairs; see [Rob14, §9.1].

2.7. Module arenas.

7 A set of morphisms S is essentially small if there is a small subset S0 ⊂ S such that every morphism in S is isomorphic to a morphism of S0 in the category Arrows(C). 2. (∞, 1)-categories 23

2.7.1. The category of arenas admits a canonical symmetric monoidal structure.

Let C1 and C2 be arenas. There is a canonical functor

C1 × C2 → C1 ⊗ C2 which commutes with small colimits in each argument, and for any arena D, the canonical functor Funct !(C1 ⊗ C2, D) → Funct(C1 × C2, D) is fully faithful with essential image spanned by functors C1 × C2 → D that commute with small colimits in each argument.

2.7.2. A symmetric monoidal arena is a commutative monoid object in the category of arenas. Equivalently, this is a symmetric monoidal category whose underlying category is an arena, and such that the monoidal product commutes with small colimits in each variable. For example, given a small category C, the category P(C) is a cartesian monoidal arena, i.e. a symmetric monoidal arena whose monoidal product is given by the cartesian product.

2.7.3. We will write Arenamon for the category of symmetric monoidal arenas.

2.7.4. By the adjoint functor theorem, the symmetric monoidal structure on an arena is auto- matically closed, i.e. there exists a bifunctor HomC(−, −) which is right adjoint to − ⊗ −.

2.7.5. Let O be a symmetric monoidal arena. An O-module arena is an O-module in the category of arenas. We let O-mod denote the category of O-module arenas. A morphism of O-modules u : M → N in particular gives rise to functorial isomorphisms o ⊗ u(n) = u(o ⊗ m) for any objects o ∈ O, m ∈ M, n ∈ N (but also contains much more coherence data).

2.7.6. One can define a notion of lax (resp. colax) functor of O-modules. This essentially amounts to requiring that, instead of isomorphisms as above, we have morphisms o ⊗ u(n) → u(o ⊗ m) (resp. u(o ⊗ m) → o ⊗ u(n)). The following lemma will be useful: Lemma 2.7.7. Let O be a symmetric monoidal arena. Let M and N be O-module arenas. Suppose that there is an adjunction u : M → N, v : N → M of underlying categories. Then the structure of colax functor of O-modules on u is equivalent to the structure of lax functor of O-modules on v.

2.8. Toposes.

2.8.1. An important subclass of arenas is formed by toposes: Definition 2.8.2. A topos is a category X such that there exists a small category C and an exact localization γ : P(C) → X. A morphism of toposes is a functor admitting an exact left adjoint.

Any topos has the property of universality of colimits:

Proposition 2.8.3. For any morphism f : x → y, the functor X/y → X/x given by the 0 0 assignment (y → y) 7→ (y ×y x → x) commutes with small colimits. 24 0. PRELIMINARIES

2.8.4. Let f : x → y be a morphism in a topos X. ˇ The Cech nerve of f is the simplicial object of X/y defined degreewise as the (n + 1)-fold fibred product

C(ˇ x/y)n = x × x × · · · × x. y y y

ˇ More precisely, the functor f 7→ C(x/y)• on the slice category X/y is right adjoint to the functor that evaluates a simplicial object at degree zero. We define: Definition 2.8.5. The morphism f : x → y is an effective epimorphism if the canonical morphism lim C(ˇ x/y) → e −→ n n∈∆op is invertible, where e denotes the terminal object of X.

2.8.6. Let X be a topos. Every morphism f : x → y is (−2)-connected. It is (−1)-connected if it is an effective epimorphism. It is n-connected if it is an effective epimorphism and its diagonal morphism x → x ×y x is (n − 1)-connected.

2.8.7. In a topos X, the classes of n-connected and n-truncated morphisms form an orthogonal factorization system, for each n > −2. In particular, every morphism f in X admits a factorization of the form f = p ◦ i, where i is n-truncated and p is n-connected. Further, we have: Lemma 2.8.8. Let X be a topos. A morphism f : x → y in X is invertible if and only if it is both n-truncated and n-connected, for some n > −2.

2.9. Sites.

2.9.1. Recall the notion of Grothendieck (pre)topologies on ordinary categories from [SGA 4]. These can be adapted to the setting of (∞, 1)-categories. In fact, a topology on a category C is equivalent to a topology on its underlying ordinary category (C)ordn. A site is a category equipped with a Grothendieck topology.

2.9.2. Given a topology τ on a small category C, let Sτ denote the (small) set of morphisms

R ,→ hC(c) for c an object of C and R ,→ hC(c) a τ-covering sieve.

A τ-sheaf on C is by definition a presheaf F which is Sτ -local. That is, for each τ-covering sieve R ,→ hC(c), the canonical morphism of spaces

(2.4) F(c) = Maps(hC(c), F) → Maps(R, F) is invertible. (We also refer to this property as τ-descent.)

We say that a morphism of presheaves is a τ-local equivalence if it is an Sτ -local equivalence.

2.9.3. We write Shτ (C) for the full subcategory of τ-sheaves. By [Chap. 0, Proposition 2.6.15] we obtain that Shτ (C) is an accessible localization of the arena P(C). In fact, this is an exact localization (hence a topos). 2. (∞, 1)-categories 25

2.9.4. If the topology τ is generated by a pretopology, then the condition of τ-descent can be phrased equivalently as follows: for every τ-covering family (fα : cα → c)α, the canonical morphism of spaces (2.5) F(c) → lim Γ(C(ˇ c /c) , F) ←− α n n∈∆ is invertible.

Here C(ˇ cα/c)• denotes the Cechˇ nerve of the morphism tαcα → c.

2.10. Stable categories.

2.10.1. Let C be a category. We say that C is pointed if it admits a zero object, i.e. an object which is both terminal and initial.

2.10.2. Let C be a pointed category admitting finite colimits and limits. For any object c in C, we can form its suspension object Σ(x) := 0 t 0, x and its loop space object Ω(x) := 0 × 0, x where 0 is a zero object. The functors (Σ, Ω) form an adjoint pair; if it is an equivalence, then we say that the category C is stable. In this case, we will write F[1] := Σ(F) and F[−1] := Ω(F).

2.10.3. Equivalently, a category C as above is stable if any commutative square in C is cartesian if and only if it is cocartesian.

2.10.4. The most fundamental example of a stable category is Spt, the category of spectra. This is obtained from the category of spaces by a formal procedure called stabilization. In particular, there is a canonical adjoint pair Σ∞ : Spc → Spt, Ω∞ : Spt → Spc.

2.10.5. There is a canonical t-structure on the category of spectra.

A spectrum E is n-connected (resp. n-connective) if πi(E) = 0 for each i 6 n (resp. for each i < n). It is n-coconnected (resp. n-coconnective) if πi(E) = 0 for each i > n (resp. for each i > n). We will abbreviate the term 0-connective (= (−1)-connected) to connective, and 0-coconnective (= 1-coconnected) to coconnective.

2.10.6. Commutative monoids in the category Spt are E∞-ring spectra. (We will not use this notion in this text.) 26 0. PRELIMINARIES

3. (∞, 2)-categories

The references for this section are [GR16] and [Lur09c].

3.1. (∞, 2)-categories as complete Segal spaces. We will use complete Segal spaces in (∞, 1)-Cat to model (∞, 2)-categories.

3.1.1. An (∞, 2)-category C is a complete Segal space in the (∞, 1)-category of (∞, 1)-categories. That is, it is the datum of a simplicial object Seq•(C) in the (∞, 1)-category of (∞, 1)-categories satisfying the following conditions:

(1) The (∞, 1)-category Seq0(C) is an ∞-groupoid. (2) The canonical functor Seq (C) → Seq (C) × Seq (C) is an equivalence. m+n m Seq0(C) n Spc (3) The Segal space (Seq•(C)) is complete (see [GR16, Chap. A.1]). We will write (∞, 2)-Cat for the (∞, 1)-category of (∞, 2)-categories.

3.1.2. There are other models for the (∞, 1)-category of (∞, 2)-categories. For example, the (∞, 1)-category (∞, 2)-Cat is equivalent to the (∞, 1)-category of 2-fold complete Segal spaces. This equivalence is induced by an equivalence between the (∞, 1)-category of (∞, 1)-categories and that of 1-fold complete Segal spaces (i.e. complete Segal spaces in the (∞, 1)-category of spaces); see [JT07]. It follows, according to [Hau15], that it is also equivalent to the (∞, 1)-category of (∞, 1)- categories enriched in (∞, 1)-Cat.

3.1.3. Any ordinary 2-category can be viewed as an (∞, 2)-category where Seq1(C) is an ordinary category. This defines a fully faithful embedding from the (∞, 1)-category of ordinary 2-categories to the (∞, 1)-category of (∞, 2)-categories. Conversely, any (∞, 2)-category gives rise to an ordinary 2-category (C)2-ordn. This defines a left adjoint to the above embedding.

3.1.4. Given an (∞, 2)-category C, there is an (∞, 2)-category (C)1-op obtained by reversing the directions of 1-morphisms, an (∞, 2)-category (C)2-op obtained by reversing the directions of 2-morphisms, and an (∞, 2)-category (C)1&2-op obtained by reversing the directions of both 1- and 2-morphisms.

3.1.5. There is an (∞, 2)-category (∞, 1)-Cat of (∞, 1)-categories, whose underlying (∞, 1)- category coincides with (∞, 1)-Cat.

3.1.6. In the sequel, the term “2-category” will mean (∞, 2)-category by default; when we need to refer to the classical notion of 2-category, we will say “(2, 2)-category”.

3.2. Passage to right/left adjoints.

3.2.1. Recall that in an 2-category C, there is a notion of adjunction between two objects x and y. A pair (f : x → y, g : y → x) forms an adjunction if and only if it defines an adjunction in the underlying ordinary 2-category (C)2-ordn. 3. (∞, 2)-categories 27

3.2.2. Let C and D be 2-categories. We say that a 2-functor u : C → D is right-adjointable (resp. left-adjointable) if, for each 1-morphism f : x → y in C, its image u(f) admits a right adjoint (resp. a left adjoint) in the 2-category D.

3.2.3. Let Maps!(S, T) denote the space of right-adjointable 2-functors. Let Maps∗(S, T) denote the space of left-adjointable 2-functors. Lemma 3.2.4. There is a canonical isomorphism of spaces 1&2-op Maps!(S, T) = Maps∗((S) , T).

Given a right-adjointable 2-functor u : S → T, we will call the corresponding functor 1&2-op u∗ :(S) → T the functor obtained from u by passage to right adjoints. Dually, given a left-adjointable 2-functor u : S → T, we will call the corresponding functor 1&2-op u! :(S) → T the functor obtained from u by passage to left adjoints.

3.3. Adjointable squares. In this section we will formulate the notion of horizontally/vertically left/right-adjointable square in a 2-category, which we will be used in the text to express base change formulas.

3.3.1. We fix an (∞, 2)-category C. Let Θ be a square in C C u C0

(3.1) v v0 0 D u D0 which commutes up to an invertible 2-morphism v0u −→∼ u0v.

Suppose that v (resp. v0) admits a right adjoint vR (resp. (v0)R) in C. Then the square Θvert:R C u C0

0 R (3.2) vR (v ) 0 D u D0 commutes up to the 2-morphism (3.3) uvR → (v0)Rv0uvR −→∼ (v0)Ru0vvR → (v0)Ru0, where the first morphism is obtained by precomposition with the counit of the adjunction, the isomorphism in the middle is given by the commutativity of the square Θ, and the final morphism is given by the unit of the adjunction. If this 2-morphism is invertible, then we say that the square Θ is vertically right-adjointable.

3.3.2. Similarly if v (resp. v0) admits a left adjoint vL (resp. (v0)L), then the square Θvert:L C u C0

0 L (3.4) vL (v ) 0 D u D0 commutes up to the 2-morphism (3.5) (v0)Lu0 → (v0)Lu0vvL ←−∼ (v0)Lv0uvL → uvL. 28 0. PRELIMINARIES

If this is invertible, we say that the square Θ is vertically left-adjointable.

3.3.3. If u (resp. u0) admits a right adjoint uR (resp. (u0)R), then the square Θhoriz:R

C C0 uR (3.6) v v0 D D (u0)R commutes up to a 2-morphism (3.7) vuR → (u0)Rv0. If it is invertible, we say that Θ is horizontally right-adjointable. Similarly, if u (resp. u0) admits a left adjoint uL (resp. (u0)L), then the square Θhoriz:L

C C0 uL (3.8) v v0 D D (u0)L commutes up to a 2-morphism (3.9) (u0)Lv0 → vuL. If it is invertible, we say that Θ is horizontally left-adjointable.

3.3.4. We have: Lemma 3.3.5. Suppose that in the square Θ (3.1), u (resp. u0) admits a right adjoint uR (resp. (u0)R), and v (resp. v0) admits a left adjoint vL (resp. (v0)L). Then Θ is vertically left-adjointable if and only if it is horizontally right-adjointable.

Proof. The square Θvert:L (resp. Θhoriz:R) commutes up to a 2-morphism α :(v0)Lu0 → uvL (resp. β : vuR → (u0)Rv0). The category of left adjoint functors C → D is equivalent to the category of right adjoint functors D → C (see [GR16, Chap. A.3, Cor. 3.1.9]), and under this equivalence the morphism α corresponds to the morphism β. 

3.4. The 2-category of arena modules.

3.4.1. Let Arenamod denote the closed symmetric monoidal (∞, 1)-category of pairs (O, C) with O a symmetric monoidal arena and C an O-module arena. We refer to [Lur16, Def. 3.3.3.8], where it is defined as an ∞-operad, [Lur16, Thm. 4.5.2.1], where it is shown to be a symmetric monoidal (∞, 1)-category, and [Lur16, Cor. 4.4.2.15], where its symmetric monoidal structure is shown to be closed.

3.4.2. There is a canonical functor (3.10) Arrows(Arenamon) → Arenamod defined on objects by the assignment (u : O → O0) 7→ (O, O0), where O0 is viewed as an O-module via the symmetric monoidal functor u. 3. (∞, 2)-categories 29

3.4.3. Let Arenamod denote the (∞, 2)-category described informally as follows: Its objects are the same as those of Arenamod, i.e. pairs (O, C) with O a symmetric monoidal arena and C an O-module arena. Its 1-morphisms (O, C) → (O0, C0) are the same as those of Arenamod, i.e. pairs (u, v) with u : O → O0 a symmetric monoidal morphism of arenas and v : C → C0 a morphism of O-module categories, where C0 is viewed as a O-module via the functor u. Its 2-morphisms (u, v) → (u0, v0) are pairs (α, β) with α : u → u0 a symmetric monoidal natural transformation, and β : v → v0 an O-linear natural transformation. We refer to [Hau14] for the precise definition as a 2-fold complete Segal space. 30 0. PRELIMINARIES

4. Derived schemes

In this section we give the definition of derived schemes, following the functorial approach of To¨en–Vezzosi [TV08] [MT10]. It is also possible to give a definition using locally ringed (∞, 1)-toposes, in line with the definition of classical schemes based on locally ringed toposes; see [Lur09a].

4.1. Simplicial commutative algebra.

4.1.1. Consider the full subcategory Poly of the category of (small) commutative rings, spanned by the polynomial algebras Z[T1,..., Tn](n > 0). Definition 4.1.2. A simplicial commutative ring is a weakly inductive presheaf of spaces on Poly.

In other words, it is a presheaf which sends finite coproducts in Poly to products; see [Chap. 0, Paragraph 2.5]. Let SCRing denote the category of simplicial commutative rings, a full subcategory of the category of presheaves on Poly. Recall from [Chap. 0, Proposition 2.5.6] that this is the free completion of Poly by sifted colimits; in particular, it is an arena. Remark 4.1.3. One can show that SCRing is the (∞, 1)-category obtained from the ordinary category of simplicial objects in the category of commutative rings, by inverting weak homotopy equivalences (i.e. taking an (∞, 1)-categorical localization). See [Lur09a, Rmk. 4.1.2].

4.1.4. There is a canonical functor A 7→ ASpc, sending a simplicial commutative ring to its underlying space.

This is defined by evaluation on the polynomial ring Z[T], i.e. ASpc := A(Z[T]). Note that this is conservative.

The space ASpc has a canonical base point pt → ASpc, induced by the canonical homomor- phism Z → Z[T]. We write H−n(A) for its nth homotopy group. There is a canonical structure of (ordinary) commutative ring on H0(A), and of H0(A)-module on each H−n(A).

4.1.5. We also have the Eilenberg–Mac Lane functor A 7→ ASpt , sending a simplicial commutative ring to its associated (connective) E∞-ring spectrum. This is obtained from the universal property of SCRing (see [Chap. 0, Proposition 2.5.6]). Hence it is characterized as the unique functor that commutes with sifted colimits and sends a polynomial ring R = Z[T1,..., Tn] to its Eilenberg–Mac Lane spectrum RSpt . In fact, it commutes with arbitrary small colimits and limits, and is conservative.

The underlying space of the spectrum ASpt coincides with ASpc. The spectrum ASpt is n connective (i.e. H (ASpt ) = 0 for n > 0).

4.1.6. Given a simplicial commutative ring A, an A-module is a (left) module over the E∞-ring spectrum ASpt . We write A-mod for the category of A-modules, which is a stable symmetric monoidal arena. −i We denote H (M) := πi(M) for each i. Remark 4.1.7. If A is an ordinary commutative ring, then the underlying (1, 1)-category of A-mod coincides with the derived category of the abelian category of A-modules.

We let A-alg denote the category of A-algebras, i.e. the slice category A\SCRing. 4. DERIVED SCHEMES 31

4.1.8. Note that if we took presheaves of sets in [Chap. 0, Definition 4.1.2], we would recover the category of (ordinary) commutative rings. In particular, the 0-truncated simplicial commutative rings are precisely ordinary commutative rings. The 0-truncation functor, left adjoint to the inclusion of ordinary commutative rings, is nothing else than the functor A 7→ H0(A).

4.1.9. Let A be a simplicial commutative ring. For any connective A-module M, we will write SymA(M) for the free A-algebra generated by M. That is, we have a canonical isomorphism ∼ MapsA-alg(SymA(M), B) −→ MapsA-mod(M, B) for each A-algebra B.

4.2. Prestacks.

4.2.1. A prestack is a presheaf of spaces on the category (SCRing)op, i.e. a functor SCRing → Spc (see [Chap. 0, Paragraph 2.6]). We write Prestk for the category of prestacks.

4.2.2. Given a simplicial commutative ring A, we will write Spec(A) for the prestack represented by A. We say that a prestack is an affine scheme if it is represented by a simplicial commutative ring, and write Schaff for the full subcategory of Prestk spanned by affine schemes. Let S be a prestack. For a simplicial commutative ring A, we say that an A-point of S is a morphism s : Spec(A) → S, or equivalently a point of the space S(A).

4.2.3. A classical8 prestack is a presheaf on the opposite of the ordinary category of commutative rings.

Given a prestack S, let Scl denote its underlying classical prestack, defined as the restriction to ordinary commutative rings. The functor S 7→ Scl admits a fully faithful left adjoint, embedding the category of classical prestacks as a full subcategory of prestacks. We refer to prestacks of the form Spec(A), with A an ordinary commutative ring, as classical affine schemes. The functor S 7→ Scl sends affine schemes to classical affine schemes: we have 0 Spec(A)cl = Spec(H (A)) for any simplicial commutative ring A.

4.3. Quasi-coherent sheaves.

4.3.1. Let S = Spec(A) be an affine scheme. A quasi-coherent module on S is the datum of an A-module. We write OSpec(A) for the quasi-coherent module given by A, viewed as a module over itself.

4.3.2. Let S be a prestack. A quasi-coherent OS-module consists of the following data: (1) For every affine scheme Spec(A) and every morphism s : Spec(A) → S, a quasi-coherent module Fs on OSpec(A).

8The adjective classical refers to the fact that they are defined on non-derived objects (ordinary commutative rings, not simplicial commutative rings). In the literature they have been studied by C. Simpson and others under the name higher prestacks, since they may take values in arbitrary spaces, not just groupoids. In our terminology, prestacks are “higher” by default, and “non-higher” prestacks are 1-truncated prestacks. 32 0. PRELIMINARIES

(2) For every pair of morphisms s : Spec(A) → S, s0 : Spec(B) → S fitting into a commutative triangle Spec(A) s S, s0 f Spec(B)

∗ ∼ an isomorphism f (Fs0 ) −→ Fs. (3) A homotopy coherent system of compatibilities between all such isomorphisms.

4.3.3. More precisely, we define the category Qcoh(S) as the limit Qcoh(S) := lim Qcoh(Spec(A)) ←− Spec(A)→S in the category of arenas. This category is stable, as the property of stability is stable under limits of (∞, 1)-categories. Better yet, we can define a presheaf of symmetric monoidal arenas S 7→ Qcoh(S) as the right Kan extension of the presheaf A 7→ A-mod along the Yoneda embedding (SCRing)op → Prestk. In particular, for each morphism of prestacks f, we have a symmetric monoidal colimit- ∗ preserving functor f , the inverse image functor, and its right adjoint f∗, the direct image functor.

4.3.4. Let S be a prestack. We write OS for the quasi-coherent module defined by OS,s = OSpec(A) for each affine scheme Spec(A) and each A-point s : Spec(A) → S. This is the unit of the symmetric monoidal structure. Given a quasi-coherent module F on S, we write Γ(X, F) for the space of sections over an ∗ S-scheme X. This is by definition the mapping space Maps(OX, p (F)), where p :X → S is the structural morphism.

4.4. Stacks.

4.4.1. Let f :T → S be a morphism of affine schemes, S = Spec(A), T = Spec(B). Definition 4.4.2. (i) The morphism f is of finite presentation if B is compact in the category of A-algebras. (ii) The morphism f is flat if the functor f ∗ : Qcoh(S) → Qcoh(T) is exact. Equivalently, 0 0 the morphism fcl : Spec(H (B)) → Spec(H (A)) of underlying classical affine schemes is flat (i.e. H0(B) is flat, in the usual sense, as an H0(A)-module), and the canonical morphism −i 0 −i H (A) ⊗H0(A) H (B) → H (B) is invertible for each i. (iii) The morphism f is an open immersion if it is a flat monomorphism9 of finite presenta- 0 0 tion. Equivalently, it is flat and the morphism fcl : Spec(H (B)) → Spec(H (A)) of underlying classical schemes is an open immersion (in the classical sense).

4.4.3. The Zariski topology on Schaff is the Grothendieck topology associated to the following pretopology. A family of morphisms of affine schemes (jα :Uα → X)α∈Λ is Zariski covering if and ∗ only if each jα is an open immersion, and the family of functors (jα) : Qcoh(X) → Qcoh(Uα) is conservative. Definition 4.4.4. A stack is a prestack satisfying descent with respect to the Zariski topology.

9 I.e. the canonical morphism T → T ×S T is invertible. 4. DERIVED SCHEMES 33

We write Stk for the full subcategory of Prestk spanned by stacks.

4.5. Schemes.

4.5.1. Let j :U → S be a morphism of stacks. Definition 4.5.2. (i) If S is affine, then j is an open immersion if it is a monomorphism, and there exists a family (jα :Uα → S)α, with each jα an open immersion of affine schemes [Chap. 0, Definition 4.4.2] that factors through U and induces an effective epimorphism tαUα → U. (ii) For general S, the morphism j is an open immersion if for each simplicial commutative ring A and each A-point s : Spec(A) → S, the base change U ×S Spec(A) → Spec(A) is an open immersion in the sense of (i).

4.5.3. Let S be a stack. A Zariski cover of S is a small family of open immersions of stacks (jα :Uα → S)α such that the canonical morphism tαUα → S is surjective (i.e. an effective epimorphism in the topos of stacks). If each Uα is an affine scheme, we call this an affine Zariski cover. We define: Definition 4.5.4. A scheme is a stack S which admits an affine Zariski cover.

We write Sch for the full subcategory of Stk spanned by schemes. It is closed under coproducts and fibred products.

Definition 4.5.5. (i) A scheme S is quasi-compact if for any Zariski cover (jα :Uα → S)α∈Λ, there exists a finite subset Λ0 ⊂ Λ such that the family (jα)α∈Λ0 is still a Zariski cover. (ii) A morphism of schemes f :T → S is quasi-compact if for any simplicial commutative ring A and any A-point s : Spec(A) → S, the scheme T ×S Spec(A) is quasi-compact. (iii) A scheme S is quasi-separated if for any open immersions U ,→ S and V ,→ S, with U and V affine, the intersection U ×S V is quasi-compact.

4.5.6. We define a classical scheme to be a Zariski sheaf of sets on the category (CRing)op, admitting a Zariski affine cover. This is equivalent to the definition of scheme given in [EGA IS].

Given a scheme S, the underlying classical prestack Scl takes values in sets, and is a classical scheme. We therefore refer to Scl as the underlying classical scheme of S.

4.6. Closed immersions.

4.6.1. Let f :Y → X be a morphism of schemes. We say that the morphism f is affine if, for any simplicial commutative ring A and A-point x : Spec(A) → X, the base change Y ×X Spec(A) is an affine scheme.

4.6.2. If X = Spec(A) and Y = Spec(B) are affine, the morphism f is a closed immersion if the homomorphism A → B induces a surjection H0(A) → H0(B). In general the morphism f is a closed immersion if it is affine, and for any simplicial commutative ring A and A-point x : Spec(A) → X, the base change Y ×X Spec(A) → Spec(A) is a closed immersion of affine schemes. Equivalently, f is a closed immersion if and only if it induces a closed immersion on underlying classical schemes. 34 0. PRELIMINARIES

4.6.3. Let i :Z ,→ S be a closed immersion of schemes. Let U be the prestack defined as follows: for a simplicial commutative ring A, its A-points are A-points s : Spec(A) → S such that the base change Spec(A) ×S Z is the empty scheme. One can show that U is a scheme, and that the canonical morphism U → S is an open immersion. We call j :U ,→ S the complementary open immersion to i.

4.7. Vector bundles.

4.7.1. Just as in [Chap. 0, Paragraph 4.3], we can define a notion of quasi-coherent algebra on a prestack S, such that the category of quasi-coherent algebras on Spec(A) coincides with the category of A-algebras.

4.7.2. Let S be a scheme and A a quasi-coherent algebra on S. Consider the presheaf SpecS(A) on the category of schemes over S, which sends an S-scheme X with structural morphism f to ∗ the space of quasi-coherent algebra homomorphisms Maps(f (A), OX).

The presheaf SpecS(A) clearly satisfies Zariski descent. Hence it defines a stack over S (there is a canonical equivalence Sh(Sch/S) = Sh(Sch)/S = Sh(Schaff )/S), which we call the relative spectrum of the quasi-coherent algebra A. In fact, we have: Lemma 4.7.3. Let A be a connective quasi-coherent algebra over a scheme S. Then the stack SpecS(A) is a scheme.

This follows from functoriality in S, and the fact that for S = Spec(A) affine, we have SpecS(A) = Spec(Γ(S, A)).

4.7.4. Let F be a connective quasi-coherent module on S. The (connective) quasi-coherent algebra SymOS (F) is defined by SymOS (F)s := SymOS,s (Fs) for each simplicial commutative ring A and A-point s : Spec(A) → S.

4.7.5. A connective quasi-coherent module F on S is locally free of rank n if there exists a Zariski ∗ cover (jα :Uα → S)α such that each inverse image jα(F) is a free quasi-coherent OSα -module of rank n, i.e. j∗ (F) = O⊕n. α Sα Given a locally free module of finite rank F, we define:

∨ Definition 4.7.6. The vector bundle associated to F is the S-scheme V(F) := SpecS(SymOS (F )).

Note that any global section s ∈ Γ(S, F) defines a section s :S ,→ V(F) of the structural morphism, which is a closed immersion. In particular, any vector bundle admits a zero section.

4.7.7. For an integer n > 0, we define the affine space of dimension n over a scheme S, denoted n ⊕n AS , to be the total space of the free OS-module OS : n ⊕n AS = V(OS ).

n n For any morphism of schemes f :T → S, we have AS ×S T = AT. Since the structural morphism An → S is flat, it follows that there is a canonical isomorphism (An) = An . S S cl Scl

1 4.7.8. The affine line AS over S is the affine space of dimension 1. Since the quasi-coherent module OS has a unit section (being a quasi-coherent algebra), the affine line admits both a zero and a unit section. 5. LOCAL PROPERTIES OF MORPHISMS 35

5. Local properties of morphisms

The reference for this section is [TV08] [MT10].

5.1. Square-zero extensions.

5.1.1. Let p :Y → X be a morphism of affine schemes, X = Spec(A), Y = Spec(B). Given a connective quasi-coherent module F on Y, we let

Y ,→ YF := Spec(B ⊕ M) denote the trivial square-zero extension of Y along F, where M = Γ(Y, F).

The morphism Y ,→ YF is the closed immersion induced by the homomorphism B ⊕ M → B, (b, m) 7→ b.

5.1.2. A derivation of Y over X with values in F, is a retraction of the morphism Y ,→ YF (in the category of affine schemes over X). There is a canonical retraction, the trivial derivation dtriv, defined by the morphism B → B ⊕ M, b 7→ (b, 0). Let Der(Y/X, F) denote the space of derivations in F.

5.1.3. Let F be a 0-connected quasi-coherent module on Y.

Any derivation d of Y/X valued in F gives rise to a square-zero extension i :Y ,→ Yd. This is the closed immersion (in fact, nil-immersion10) defined as the cobase change of the trivial derivation along d, so that there is a cocartesian square

dtriv YF Y

(5.1) d i Y Yd in the category of affine schemes.

5.1.4. The following important fact reduces many proofs in derived algebraic geometry to their classical analogues, by induction along square-zero extensions. Proposition 5.1.5. Let S = Spec(A) be an affine scheme. Then there exists a sequence of nil-immersions of affine schemes 0 −1 −n (5.2) Scl = S> ,→ S> ,→ · · · ,→ S> ,→ · · · ,→ S, with S>−n = Spec(A>−n), satisfying the following properties: −n −n (i) For each n > 0, the homomorphism A → A> identifies A> as the n-truncation of the simplicial commutative ring A. (ii) The sequence is functorial in A.

(iii) The canonical morphism A → lim A>−n is invertible. ←−n>0 −n −n−1 (iv) Each morphism S> ,→ S> (n > 0) is a square-zero extension by a derivation −n valued in H (OS)[n + 1]. Further, this sequence is uniquely characterized, up to isomorphism of diagrams indexed on the poset of nonnegative integers, by the property (i).

The sequence (5.2) is often called the Postnikov tower of A.

10See [Chap. 0, Paragraph 5.5]. 36 0. PRELIMINARIES

5.2. The cotangent sheaf.

5.2.1. Let p :Y → X be a morphism of affine schemes. We have: Proposition 5.2.2. The functor F 7→ Der(Y/X, F) is representable by a connective quasi- ∗ coherent module TY/X on Y.

∗ The quasi-coherent module TY/X on Y is called the (relative) cotangent sheaf of the morphism ∗ p :Y → X. We obtain the absolute cotangent sheaf TS by taking the relative cotangent sheaf of the morphism S → Spec(Z).

5.2.3. The following fact is crucial.

g f Lemma 5.2.4. Let Z −→ Y −→ X be a sequence of morphisms of affine schemes. Then there is a canonical exact triangle ∗ ∗ ∗ ∗ (5.3) g (TY/X) → TZ/X → TZ/Y of quasi-coherent sheaves on Z.

∗ In particular, we see that the relative cotangent sheaf TY/X is the cofibre of the canonical ∗ ∗ ∗ morphism f (TX) → TY.

5.2.5. The cotangent sheaf of a vector bundle has a particularly simple description: Lemma 5.2.6. Let S be an affine scheme. For any connective quasi-coherent module F on S, we have a canonical isomorphism ∗ ∼ ∗ TSpec (Sym (F))/S −→ p (F), S OS where p denotes the structural morphism of the S-scheme SpecS(SymOS (F)).

5.2.7. Let f :Y → X be a morphism of affine schemes, A a simplicial commutative ring, and y : Spec(A) → Y an A-point. Let f(y) denote the induced A-point Spec(A) → Y → X. The differential of f at y is the canonical morphism of quasi-coherent sheaves on Spec(A) ∗ ∗ (5.4) dfy : TX,f(y) → TY,y.

5.2.8. Let S be an affine scheme and let f :S → A1 be a morphism, defining a global section f ∈ Γ(S, OS). For any simplicial commutative ring A and A-point s : Spec(A) → S, the differential of f at ∗ s defines (by [Chap. 0, Lemma 5.2.6]) a point dfs of (the underlying space of) TX,x.

5.2.9. Let p :Y → X be a morphism of schemes. For any simplicial commutative ring A, A-point y : Spec(A) → Y, and 0-connected quasi-coherent module F on Y, a derivation at y of p with values in F is a commutative triangle Spec(A) y

d Spec(A)F Y in the category of X-schemes.

We write Dery(Y/X, F) for the space of derivations at y. 5. LOCAL PROPERTIES OF MORPHISMS 37

5.2.10. The functor F 7→ Dery(Y/X, F) is represented by a connective quasi-coherent module ∗ TY/X,y on Spec(A), called the relative cotangent sheaf of p at y: ∼ ∗ Dery(Y/X, F) −→ MapsQcoh(Spec(A))(TY/X,y, F).

5.2.11. Given a simplicial commutative ring A and an A-point y : Spec(A) → Y, a simplicial commutative ring B and a B-point y0 : Spec(B) → Y, and a morphism of affine schemes f : Spec(B) → Spec(A) such that y ◦ f = y0, we obtain a canonical morphism of quasi-coherent modules on Spec(B) ∗ ∗ ∗ f (TY/X,y) → TY/X,y0 which is invertible. ∗ Moreover, the data of the quasi-coherent modules TY/X,y, as y varies over A-points of Y (with A an arbitrary simplicial commutative ring), together with the above isomorphisms, is compatible in a homotopy coherent way, and can therefore be refined to a connective quasi-coherent module ∗ TY/X defined on the scheme Y.

5.3. Smooth and ´etalemorphisms.

5.3.1. Let p :Y → X be a morphism of affine schemes. Definition 5.3.2. (i) The morphism p is ´etaleif it is of finite presentation and the cotangent ∗ sheaf TY/X is zero. Equivalently, p is flat and the underlying morphism of classical schemes pcl :Ycl → Xcl is ´etalein the sense of [EGA IV4]. ∗ (ii) The morphism p is smooth if it is of finite presentation and the complex sheaf TY/X is locally free of finite rank. Equivalently, p is flat and the underlying morphism of classical schemes pcl :Ycl → Xcl is smooth in the sense of [EGA IV4].

In general, for a morphism of schemes p :Y → X, we define ´etalenessand smoothness Zariski-locally on the source. That is: Definition 5.3.3. The morphism p is ´etale(resp. smooth, flat, locally of finite presentation) if there exists affine Zariski covers (Yα ,→ Y)α and (Xβ ,→ X)β together with the data of, for each α, an index β and a morphism of affine schemes Yα ,→ Xβ which is ´etale(resp. smooth, flat, locally of finite presentation) and fits in a commutative square

Yα Xβ

Y X.

Note that open immersions are ´etale,and ´etalemorphisms are smooth. By [Chap. 0, Lemma 5.2.6], any vector bundle E → S is smooth. In particular, affine spaces (and hence also projective spaces) are smooth.

5.3.4. The following, a derived version of [EGA IV4, Thm. 17.11.4], is standard: Proposition 5.3.5. A morphism p :Y → X is smooth if and only if, Zariski-locally on Y, there exists a factorization of p as a composite q (5.5) Y −→ An × X −→r X for some integer n > 0, where q is ´etaleand r is the canonical projection. 38 0. PRELIMINARIES

Proof. Since pcl :Ycl → Xcl is smooth, we can find, Zariski-locally on Ycl, sections (fi)i=1,...,n of O whose differentials d (f ) form a basis of the locally free sheaf T∗ (see the proof Ycl Ycl/Xcl i Ycl/Xcl of [EGA IV4, Thm. 17.11.4]). Choosing lifts of these sections to OY, we obtain a morphism q :Y → An × X such that the morphism ∗ ∗ ∗ q (TAn×X/X) → TY/X is invertible, and hence q is ´etaleby the exact triangle (5.3). 

5.4. Deformation along square-zero extensions.

0 ∗ 5.4.1. Let S be an affine scheme and S a square-zero extension of S by a derivation d : TS → F, for some 0-connected quasi-coherent module F. Let X be an affine scheme over S with structural morphism p. Definition 5.4.2. A deformation of X along the square-zero extension S ,→ S0 is an affine 0 0 0 scheme X over S together with an isomorphism X → X ×S0 S.

In other words, a deformation of X is a cartesian square: X X0

S S0

5.4.3. We have: Lemma 5.4.4. The datum of a deformation of X along S ,→ S0 is equivalent to the datum of a null-homotopy of the composite ∗ ∗ ∗ ∗ TX/S[−1] → p (TS) → p (F).

0 ∗ ∗ 0 Given such a null-homotopy, one obtains a derivation d : TX → p (F); the deformation X is constructed as the square-zero extension of X along d0.

∗ ∗ 5.4.5. For example, if p is smooth, then any morphism TX/S[−1] → p (F) must be null-homotopic; hence X admits a deformation along any square-zero extension S ,→ S0. If p is further ´etale, then this deformation is unique.

5.5. Cobase change along nil-immersions.

5.5.1. A nil-immersion is a closed immersion i :X0 ,→ X which induces an isomorphism (X0)cl → Xcl on underlying classical schemes.

5.5.2. Let i :X0 ,→ X be a nil-immersion and f :X0 → Y0 a morphism of schemes. We have: Lemma 5.5.3. (i) The cobase change of f along the nil-immersion i is representable in the category of schemes. That is, there exists a scheme Y fitting in the cocartesian square i X0 X

f

Y0 Y.

(ii) If f is affine then the morphism Y0 → Y is a closed nil-immersion. (iii) If f is an open immersion, then so is the morphism X → Y. 6. GLOBAL PROPERTIES OF MORPHISMS 39

5.6. Lifting smooth morphisms along closed immersions.

5.6.1. The following is a derived version of [EGA IV4, Prop. 18.1.1]: Proposition 5.6.2. Let i :Z ,→ S be a closed immersion of schemes. For any smooth (resp. ´etale) morphism p :X → Z, there exists, Zariski-locally on X, a smooth (resp. ´etale)morphism q :Y → S, and a cartesian square X Y p q Z S.

Proof. First we consider the ´etalecase. The question being Zariski-local, we may assume that S, Z and X are affine. Consider the Postnikov towers ([Chap. 0, Proposition 5.1.5]) 0 −1 −n Scl = S> ,→ S> ,→ · · · ,→ S> ,→ · · · ,→ S 0 −1 −n Zcl = Z> ,→ Z> ,→ · · · ,→ Z> ,→ · · · ,→ Z for S and Z, respectively. Since p is flat, the Postnikov tower for X is identified with the base change of the Postnikov tower of Z.

For a fixed integer n > 0, consider the following claim: (∗) There exists, Zariski-locally on X>−n, an ´etalemorphism q>−n :Y>−n → S>−n and a cartesian square X>−n Y>−n

p>−n q>−n

Z>−n S>−n.

Note that it suffices to show that (∗) holds for each n > 0, since we can conclude by passing to filtered colimits. For n = 0, the claim is [EGA IV4, Prop. 18.1.1]. We proceed by induction; assume that the claim holds for a fixed n. We define Y>−n−1 to be the deformation of Y>−n along the square-zero extension S>−n ,→ S>−n−1, which exists by [Chap. 0, Lemma 5.4.4]. Note that X>−n−1 is itself a deformation of X>−n along the square- zero extension Z>−n ,→ Z>−n−1. That the resulting square is cartesian is a straightforward verification. For the smooth case, the claim follows from the ´etale case and from [Chap. 0, Proposi- tion 5.3.5]. 

6. Global properties of morphisms

The reference for this section is [GR16, Chap. II.2].

6.1. Proper morphisms.

6.1.1. Let f :Y → X be a morphism of schemes. We say that f is of finite type if the underlying morphism of classical schemes fcl is of finite type in the sense of [EGA IS].

We say that f is separated if the diagonal morphism Y → Y ×X Y is a closed immersion. Equivalently, the underlying morphism of classical schemes fcl is separated in the sense of [EGA IS].

The morphism f is proper if the induced morphism of underlying classical schemes fcl : Ycl → Xcl is proper in the sense of [EGA II]. 40 0. PRELIMINARIES

6.1.2. Let E be a locally free module of finite rank over a scheme S. Consider the presheaf on the category of S-schemes sending an S-scheme X with structural morphism f to the space of direct summands of f ∗(E)∨ which are locally free of rank 1. This satisfies Zariski descent and corresponds to a stack over S which we denote P(E) and call the projective bundle associated to E. One can construct an affine Zariski cover of P(E), so that it is in fact a scheme.

Further, since the structural morphism P(E) → S is flat, we have P(E)cl = P(E) ×S Scl = ∗ P(i E), where i :Scl ,→ S is the canonical inclusion. For classical schemes, our construction ∗ coincides with that of [EGA IS] by definition, and in [EGA II] it is proved that P(i E) → Scl is proper. Hence we obtain: Lemma 6.1.3. The stack P(E) is a scheme, and the structural morphism p : P(E) → S is proper.

⊕n+1 6.1.4. Taking E = OS , we obtain the projective space of dimension n over S: n ⊕n+1 PS := P(OS ) for each n > 0. n In this case, an affine Zariski cover can be chosen of the form (Ui ,→ PS )i=0,...,n where for n each i there is an isomorphism Ui = AS . Also, [Chap. 0, Lemma 6.1.3] can be deduced in a less direct way from the following n observations: (1) for S = Spec(Z), PS coincides with the classical proper S-scheme constructed n n in [EGA IS]; (2) the construction is functorial, i.e. PS ×S T = PT for any morphism T → S; (3) the structural morphism Pn → S is flat, so that (Pn) = Pn × S = Pn . S S cl S S cl Scl

1 6.1.5. In particular, the projective line PS fits into a cartesian and cocartesian square of schemes

1 × 1 (AS) AS

1 1 AS PS

1 × 1 where (AS) denotes the complement of the zero section S ,→ A .

6.2. Closure.

6.2.1. Let f :Y → X be a morphism of schemes. Let Closed(f) denote the category of factorizations f Y X i

Z with i a closed immersion. We have: Lemma 6.2.2. The category Closed(f) admits an initial object.

This initial object gives in particular a closed immersion i : f(Y) ,→ X, which is called the closure of f. If f is a closed immersion, then the canonical morphism Y → f(Y) is invertible. 6. GLOBAL PROPERTIES OF MORPHISMS 41

g f 6.2.3. Let X00 −→ X0 −→ X be a sequence of morphisms of schemes. We have the following transitivity property of closure: Lemma 6.2.4. The canonical morphism f ◦ g(X00) → f 0(g(X00)) is invertible, where f 0 : g(X00) → X is the restriction of f.

6.3. Compactifications of morphisms.

6.3.1. Recall the following definition from [SGA 4, Exp. XVII, 3.2.5]. Definition 6.3.2. Let f :Y → X be a morphism of schemes. A compactification of f is a commutative triangle f Y X j f 0

Y0 where j is an open immersion and f 0 is proper.

A morphism of compactifications is a morphism of commutative triangles. We write Compact(f) for the category of compactifications of f.

6.3.3. In the setting of classical schemes, the category of compactifications Compact(f) is cofiltered [SGA 4, Exp. XVII, Prop. 3.2.6], and hence contractible, for any separated and of finite type morphism f between quasi-compact quasi-separated schemes. In the derived setting, we have the following statement, demonstrated in [GR16, Book-II.2, Prop. 2.1.6]: Proposition 6.3.4. Let f :Y → X be a separated morphism of finite type between quasi-compact quasi-separated schemes. Then the category Compact(f) is contractible.

Proof. The proof of loc. cit. applies mutatis mutandis. We briefly recall the argument here. First, one considers the full subcategory of dense compactifications, for which the open immersion j :Y ,→ Y0 induces an isomorphism j(Y) → Y0 on the closure. Given any compacti- j fication (Y −→ Y0 → X), we can form a dense compactification by replacing j with Y → j(Y). This provides a right adjoint to the inclusion of this subcategory, so it suffices to show that this subcategory is contractible11. By the classical Nagata compactification theorem, as generalized to quasi-compact quasi- separated schemes by Deligne [Con07], the morphism fcl :Ycl → Xcl admits a compactification 0 0 0 (Ycl ,→ Ycl → Xcl). We define Y to be the cobase change of Ycl along the nil-immersion 0 Ycl → Y. This exists by [Chap. 0, Lemma 5.5.3] and defines a compactification (Y ,→ Y → X) of f. Taking the associated dense compactification, this shows that the category of dense compactifications is not empty. Hence it suffices to show that it admits binary products. 0 0 For this we take two dense compactifications (Y ,→ Y1 → X) and (Y ,→ Y2 → X). Their product is given by the dense compactification defined by the closure of Y inside the fibred 0 0 product Y1 ×X Y2. 

11Recall that an adjunction of (∞, 1)-categories induces isomorphisms on underlying ∞-groupoids.

CHAPTER 1

Motivic spaces and spectra

1. Introduction 44 1.1. Motivic spaces 44 1.2. Motivic spectra 44 1.3. The localization theorem 45 1.4. The exceptional operations for closed immersions 46 1.5. Organization of this chapter 46 2. Motivic spaces 48 2.1. Fibred spaces 48 2.2. Nisnevich descent 48 2.3. A1-homotopy invariance 50 2.4. Motivic spaces 51 3. Pointed motivic spaces 53 3.1. Pointed fibred spaces 53 3.2. Pointed motivic spaces 54 4. Motivic spectra 55 4.1. Fibred spectra 55 4.2. Motivic spectra 56 5. Inverse and direct image functoriality 59 5.1. For motivic spaces 59 5.2. For pointed motivic spaces 59 5.3. For motivic spectra 60 6. Smooth morphisms 61 6.1. The functor p] 61 6.2. Smooth base change formulas 62 6.3. Smooth projection formulas 63 7. Closed immersions 64 7.1. A topological digression, I: local cocontinuity 64 7.2. A topological digression, II: contractible colimits 66 7.3. The exceptional inverse image functor i! 67 7.4. The localization theorem 68 7.5. Closed base change formula 69 7.6. Closed projection formula 70 7.7. Smooth-closed base change formula 71 8. Thom spaces 72 8.1. Thom spaces 72 8.2. The Thom space of A1 72 8.3. Motivic P1-spectra 73 9. The localization theorem 74 9.1. The space hS(X, t) 74 9.2. Etale´ base change 76 9.3. Reduction to the case of vector bundles 78 9.4. Motivic contractibility of hS(X, t) 80 9.5. Proof of the localization theorem 81

43 44 1. MOTIVIC SPACES AND SPECTRA

1. Introduction

1.1. Motivic spaces.

1.1.1. Let S be a derived scheme. A motivic space over S is a presheaf of spaces F on the category of smooth derived S-schemes, satisfying the properties of Nisnevich descent and A1-homotopy invariance.

1.1.2. Nisnevich descent can be formulated equivalently as the following excision property (see [Chap. 1, Proposition 2.2.6]): Consider a Nisnevich square, i.e. a cartesian square of smooth derived S-schemes

W V (1.1) p j U X with j an open immersion and p ´etale,and the induced morphism p−1(X − U) → X − U an isomorphism of underlying reduced classical schemes. Then the induced commutative square of spaces j∗ Γ(X, F) Γ(U, F)

p∗ Γ(V, F) Γ(W, F) is cartesian.

1.1.3. Homotopy invariance is the condition that, for any smooth derived S-scheme X, the canonical morphism of spaces Γ(X, F) → Γ(A1 × X, F) is invertible.

1.1.4. Let H(S) denote the category of motivic spaces over S. It is easy to see that the assignment S 7→ H(S) admits the following functorialities: Given a morphism f :T → S of derived schemes, there is an inverse image functor f ∗ : H(S) → H(T) which is left adjoint to a direct image functor

f∗ : H(T) → H(S).

1.1.5. If f is smooth, there is a “bonus” operation

f] : H(T) → H(S), left adjoint to f ∗. ∗ This operation is compatible with the operations (f , f∗, ⊗, Hom) in the sense that it satisfies various base change and projection formulas; see [Chap. 1, Sect. 6].

1.2. Motivic spectra. 1. INTRODUCTION 45

1.2.1. Choose a family of pointed motivic spaces TS over each derived scheme S, together with isomorphisms ∗ ∼ f (TS) −→ TT for each morphism of derived schemes f :T → S (and a homotopy coherent system of compati- bilities between these isomorphisms). 1 For example, we can take the family of motivic spaces represented by projective lines PS, 1 pointed at ∞. These are isomorphic to the Thom spaces ThS(AS) (see [Chap. 1, Sect. 8]).

1.2.2. A motivic T-spectrum over S is the data of a sequence of pointed motivic spaces (Fn)n>0 over S, and isomorphisms ∼ αn : Fn −→ ΩT(Fn+1) for each integer n > 0, together with a homotopy coherent system of compatibilities between these isomorphisms. Here F 7→ ΩT(F) denotes the T-loop space functor.

1.2.3. The category SHTS (S) of motivic TS-spectra can be described as the result of formally inverting the object TS with respect to the monoidal product, in the sense of [Rob15]. In particular, it admits a canonical symmetric monoidal structure, and there is a canonical symmetric monoidal functor ∞ ΣS : H(S)• → SHTS (S) which has the universal property of being initial in the category of symmetric monoidal functors which send TS to an invertible object.

∗ 1.2.4. Our functorialities (f , f∗, f]) extend to the assignment S 7→ SHTS (S). Hence we have ∗ SH inverse image functors fSH for any morphism of schemes f, which admit right adjoints f∗ SH (resp. left adjoints f] , when f is smooth).

1.3. The localization theorem.

1.3.1. The main result of this chapter can be stated as follows. Theorem 1.3.2 (Localization). Let i :Z ,→ S be a closed immersion of derived schemes with quasi-compact open complement j :U ,→ S. Then the following statements hold: (1) For any motivic space F over S, there is a canonical cocartesian square

∗ j]j (F) F (1.2)

∗ ∗ j]j (eS) i∗i (F), where eS denotes the terminal motivic space over S. (2) For any pointed motivic space (F, x) over S, there is a canonical cofibre sequence ∗ ∗ (1.3) j]j (F, x) → (F, x) → i∗i (F, x).

(3) For any motivic TS-spectrum E, there is a canonical exact triangle ∗ ∗ (1.4) j]j (E) → E → i∗i (E).

This is the combination of [Chap. 1, Theorem 7.4.3], [Chap. 1, Corollary 7.4.5], and [Chap. 1, Corollary 7.4.7]. 46 1. MOTIVIC SPACES AND SPECTRA

1.3.3. Over classical schemes (noetherian of finite Krull dimension), this theorem was demon- strated by Morel–Voevodsky (see [MV99]). Our proof has the same general flavour, but we make some parts of the argument more robust, so that it survives in the derived setting. When we take a closed immersion i :Z ,→ S of classical schemes, we recover a proof of the localization property for classical schemes without any noetherian or finite dimensional assumptions. The point is that over a general base, our definition is not equivalent to the definition of Morel–Voevodsky. We only impose descent with respect to Cechˇ covers, instead of arbitrary hypercovers. The latter condition, called hyperdescent, is a priori much stronger, except over noetherian finite-dimensional schemes, where they coincide. We refer to [Lur11] and [Hoy15, Appendix C] for an explanation of this distinction.

1.3.4. Since the underlying classical scheme of a derived scheme is a closed subscheme with empty complement, an immediate consequence of this result is that the unstable and stable motivic homotopy categories of a derived scheme coincide with those of its underlying classical scheme.

Corollary 1.3.5 (Topological invariance). Let S be a derived scheme, and write i :Scl ,→ S for the inclusion of the underlying classical scheme. Then the adjunctions i∗ : H (S) → H (S ), iH• : H (S ) → H (S), H• • • cl ∗ • cl • ∗ SH iSH : SHTS (S) → SHTS (Scl), i∗ : SHTS (Scl) → SHTS (S) are equivalences of (∞, 1)-categories.

1.4. The exceptional operations for closed immersions.

! 1.4.1. In Chapter 2, we will construct the exceptional functorialities (f!, f ) on the categories SH(S) (together with the full formalism of six operations).

In the case of closed immersions i, the exceptional direct image i! coincides with i∗ by definition. We will prove (see [Chap. 1, Corollary 7.3.3]) that this admits a right adjoint i!, the exceptional inverse image functor.

! 1.4.2. Using the localization theorem, we will verify that these operations (i!, i ) satisfy all ∗ the desired compatibilities with the other operations (f , f∗, ⊗, Hom), i.e. base change and projection formulas. See [Chap. 1, Paragraph 7.5], [Chap. 1, Paragraph 7.6], and [Chap. 1, Paragraph 7.7]. These will be used in a critical way in Chapter 2 to obtain the formalism of six operations.

1.5. Organization of this chapter. In Sections 1-4, we construct the categories of motivic spaces and spectra, and study their basic properties. In applications it is useful to consider T-spectra with respect to general T, though we will obtain the full formalism of six operations in the case T = P1. ∗ Section 5 deals with the operations f (inverse image) and f∗ (direct image). In Section 6 we consider an extra operation p], not included in the “six operations”, associated to smooth morphisms p. In Section 7 we construct the exceptional inverse image functor i! for closed immersions. We also state the Morel–Voevodsky localization theorem and deduce some consequences. In Section 8 we introduce the operation of Thom (de)suspension with respect to a vector bundle E. We show that the Thom space of the affine line is P1. 1. INTRODUCTION 47

Section 9 deals with the proof of the localization theorem. 48 1. MOTIVIC SPACES AND SPECTRA

2. Motivic spaces

2.1. Fibred spaces.

1 2.1.1. Let S be a derived scheme and write Sm/S for the category of smooth schemes (of finite presentation) over S. Throughout the text, we redefine “smooth” as “smooth of finite presentation”.

A Sm-fibred space over S is the datum of a presheaf of spaces on the category Sm/S. We will abbreviate this to “fibred space” or even “space” when there is no risk of confusion.2 Given a space F over S, we will write Γ(X, F) for the space of sections over a smooth S-scheme X.

2.1.2. Let Spc(S) denote the category of spaces over S. Recall the following from [Chap. 0, Sect. 2]: The category Spc(S) is a topos, hence a fortiori a cartesian monoidal arena3 with the property of universality of colimits.

Every smooth S-scheme X represents a space hS(X), with Γ(Y, hS(X)) = MapsS(Y, X) for each smooth S-scheme Y. The assignment X 7→ hS(X) defines a fully faithful functor Sm/S → Spc(S) (the Yoneda embedding) and induces by pre-composition a canonical equivalence ∼ (2.1) Funct !(Spc(S), D) −→Funct(Sm/S, D) for each cocomplete category D. In other words the category Spc(S) is freely generated under colimits by the representable spaces.

Spc Spc 2.1.3. We will denote by eS := eS (resp. ∅S := ∅S ) the terminal object (resp. initial object). Spc We write ×S for the monoidal product on Spc(S), whose unit object 1S is the terminal Spc object eS. We denote the internal hom by HomS := HomS .

2.2. Nisnevich descent.

2.2.1. A Nisnevich square over S is a cartesian square of smooth S-schemes

U ×X V V (2.2) p j U X such that j is an open immersion, p is ´etale,and there exists a closed immersion Z ,→ X complementary to j such that the induced morphism p−1(Z) → Z is invertible.

The Nisnevich topology on the category Sm/S is generated by (i) the empty family over the empty scheme ∅; and (ii) families {j, p} over a smooth S-scheme X, where j and p form a Nisnevich square as in (2.2).

1As per our conventions we will omit the adjective “derived” in the sequel. 2The explanation for this terminology is the equivalence, provided by the Grothendieck construction (“straightening/unstraightening”), between presheaves on Sm/S and cartesian fibrations in spaces over Sm/S. 3Recall that by the term arena, we mean locally presentable (∞, 1)-category; a morphism of arenas is a functor that commutes with colimits. See [Chap. 0, Paragraph 2.6]. 2. MOTIVIC SPACES 49

2.2.2. Recall from [Chap. 0, Paragraph 2.9] that a fibred space F satisfies Nisnevich descent, or is Nisnevich-local, if for each Nisnevich covering family (fα :Xα → X)α, the canonical morphism of spaces (2.3) Γ(X, F) → lim Γ(C(Xˇ /X) , F) ←− α n n∈∆ is invertible, where Cˇ(Xα/X)• denotes the Cechˇ nerve of the morphism tαXα → X. Equivalently, for every Nisnevich covering sieve R ,→ hS(X), the canonical morphism (2.4) Γ(X, F) → Maps(R, F) is invertible.

2.2.3. A morphism F → F0 of spaces over S is a Nisnevich-local equivalence if the induced morphism 0 LNis(F) → LNis(F ) is invertible, or equivalently if for every Nisnevich-local space G, the canonical morphism of spaces Maps(G, F) → Maps(G, F0) is invertible.

We will write SpcNis(S) for the topos of Nisnevich-local spaces. This is an exact localization of Spc(S), i.e. the inclusion admits an exact left adjoint

LNis : Spc(S) → SpcNis(S).

2.2.4. We say that a fibred space F satisfies Nisnevich excision if (i) the space Γ(∅, F) is contractible; and (ii) for all smooth S-schemes X and all Nisnevich squares of the form (2.2), the commutative square of spaces Γ(X, F) Γ(U, F) (2.5)

Γ(V, F) Γ(U ×X V, F) is cartesian. Note that the property of Nisnevich excision is stable by filtered colimits (since filtered colimits commute with finite limits in any topos).

2.2.5. By [AHW15, Thm. 3.2.5]4, we have: Proposition 2.2.6. Let F be a fibred space over a scheme S. Then F satisfies Nisnevich descent if and only if it satisfies Nisnevich excision.

2.2.7. The following lemma follows from [Lur09b, Prop. 5.5.8.10, (3)]:

Lemma 2.2.8. Let (Xα)α be a finite family of smooth S-schemes. Then the canonical morphism of spaces over S t hS(Xα) → hS(t Xα) α α is invertible.

By [Lur09b, Lem. 5.5.8.14] it follows that the category SpcNis(S) is generated under sifted colimits by the representables. In fact, we can say even more:

4The statement is in the language of model categories, but the proof works mutatis mutandis in the setting of (∞, 1)-categories. 50 1. MOTIVIC SPACES AND SPECTRA

Proposition 2.2.9. Let S be a scheme. Then the category SpcNis(S) is generated under sifted 5 colimits by the representable spaces hS(X), where X is a smooth S-scheme which is affine .

Proof. Let hS(X) be a representable space over S. Since hS(X) satisfies Nisnevich descent, we can assume X is separated over S, by choosing an affine Zariski cover of X where the pairwise intersections are separated. Then we repeat the same argument to assume X is affine, by choosing an affine cover where the pairwise intersections are affine. 

2.3. A1-homotopy invariance.

2.3.1. A space F over S is A1-homotopy invariant if for every smooth S-scheme X, the canonical morphism of spaces Γ(X, F) → Γ(A1 × X, F) is invertible. Here A1 denotes the affine line Spec(Z[t]) as usual. 1 Let SpcA1 (S) denote the full subcategory of Spc(S) spanned by A -homotopy invariant spaces over S. This is an accessible localization at a small set of morphisms, and in particular the inclusion admits a left adjoint

(2.6) LA1 : Spc(S) → SpcA1 (S) called the A1-localization functor. Note that the property of A1-homotopy invariance is stable by small colimits, since colimits of presheaves are computed sectionwise. In particular, the inclusion SpcA1 (S) ,→ Spc(S) also admits a right adjoint.

2.3.2. A morphism F → F0 of spaces over S is an A1-homotopy equivalence if the induced morphism 0 LA1 (F) → LA1 (F ) is invertible, or equivalently if for every A1-homotopy invariant space G, the canonical morphism of spaces Maps(G, F) → Maps(G, F0) is invertible. 1 Let i0 (resp. i1) denote the zero section (resp. unit section) S ,→ A × S. Given two 1 morphisms f, g : F ⇒ G of spaces over S, an elementary A -homotopy from f to g is a morphism 1 ϕ : hS(A × S) × F → G such that the restriction to F = hS(S) × F along i0 (resp. i1) coincides with f (resp. g). We say that f and g are A1-homotopic if there exists a sequence of elementary A1-homotopies connecting them. Note that in this case the induced morphisms LA1 (F) ⇒ LA1 (G) coincide. A morphism f : F → G of spaces over S is called a strict A1-homotopy equivalence if there exists a morphism g : G → F such that the composites f ◦ g and g ◦ f are A1-homotopic to the identities. Note that strict A1-homotopy equivalences are A1-homotopy equivalences.

5In an absolute sense, i.e. affine over Spec(Z). 2. MOTIVIC SPACES 51

• 1 2.3.3. Let ∆S denote the cosimplicial fibred space over S associated to the interval AS, in the n n sense of [MV99, §2.3]. Hence ∆S ≈ AS for each n. Then we have: Proposition 2.3.4. (i) For each space F over S there is a canonical identification n (2.7) L 1 (F) = lim Hom (h (∆ ), F) A −→ S S S [n]∈∆op and in particular, for each smooth S-scheme X, there are canonical isomorphisms of spaces n Γ(X, LA1 (F)) = lim Γ(∆S × X, F). −→ S [n]∈∆op

(ii) The category SpcA1 (S) has universality of colimits. 1 (iii) The A -localization functor LA1 commutes with finite products.

Proof. The proof of (i) is just as in [MV99, §2.3]. For claim (ii), it suffices to show that the 1 A -localization functor LA1 is locally cartesian, i.e. the canonical morphism 0 0 LA1 (F × G) → F × LA1 (G) F F is invertible for any morphism of A1-homotopy invariant spaces F0 → F and any morphism G → F. This follows immediately from (i). Claim (iii) follows from the fact that the colimit in (i) is sifted (as sifted colimits commute with finite products).  Remark 2.3.5. From [Hoy17, Prop. 3.3] one deduces a version of the above proposition where the colimit in (i) is indexed on the sifted category of compositions of morphisms of the form A1 × X → X. This version is in fact sufficient for our purposes.

2.4. Motivic spaces.

2.4.1. We say that a space F over S is motivic if it is Nisnevich-local and A1-homotopy invariant.

2.4.2. We denote by H(S) the full subcategory of Spc(S) spanned by motivic spaces over S. This is an accessible localization of Spc(S): the inclusion admits a left adjoint

Lmot : Spc(S) → H(S) which can be described as the transfinite composite ◦n (2.8) L (F) = lim(L 1 ◦ L ) (F). mot −→ A Nis n>0 This follows from the fact that the properties of Nisnevich-locality and A1-homotopy invariance are stable by filtered colimits. In particular, H(S) is an arena.

2.4.3. We will say that a morphism F0 → F of spaces over S is a motivic equivalence if the induced morphism 0 Lmot(F ) → Lmot(F) is invertible, or equivalently if for each motivic space G, the induced morphism of spaces Maps(G, F0) → Maps(G, F) is invertible. We will say that a space F over S is motivically contractible if the canonical morphism F → eS is a motivic equivalence, i.e. the motivic localization Lmot(F) is a fibred space. 52 1. MOTIVIC SPACES AND SPECTRA

2.4.4. Let C be an arena. We say that a functor H : Sm/S → C satisfies Nisnevich excision if it sends any Nisnevich square, over a smooth S-scheme X, to a cocartesian square in C. (One can also define the property of Nisnevich descent for H, in a similar way.) We say that H satisfies A1-homotopy invariance if for each smooth S-scheme X, the canonical morphism H(X × A1) → H(X) is invertible in C.

Let Funct mot(Sm/S, C) denote the full subcategory of Funct(Sm/S, C) spanned by functors that satisfy Nisnevich excision (or equivalently, Nisnevich descent) and A1-homotopy invariance. We have the following universal property for the category H(S): Theorem 2.4.5. Let C be an arena. For any scheme S, the canonical functor

(2.9) Funct !(H(S), C) → Funct mot(Sm/S, C), given by restriction along the functor Sm/S → H(S), is an equivalence of categories.

Proof. Let Funct !,mot(Spc(S), C) denote the full subcategory of Funct !(Spc(S), C) spanned by functors that send motivic equivalences to isomorphisms in C. Under the equivalence (see [Chap. 0, Proposition 2.3.7]) ∼ Funct !(Spc(S), C) −→Funct(Sm/S, C), it is clear that the full subcategory Funct !,mot(Spc(S), C) identifies with Funct mot(Sm/S, C). Hence by [Chap. 0, Proposition 2.6.17] we have canonical equivalences ∼ ∼ Funct !(H(S), C) −→Funct !,mot(Spc(S), C) −→Funct mot(Sm/S, C) as claimed. 

2.4.6. We have:

Lemma 2.4.7. The localization functor Lmot commutes with finite products.

Proof. This follows from the formula (2.8): the functors LNis and LA1 both commute with finite products, and filtered colimits commute with finite products in the topos Spc(S). 

By adjunction it follows that for a motivic space F, the internal hom space HomS(G, F) is motivic for any space G. In particular: Corollary 2.4.8. The category H(S) is cartesian closed.

In particular we get a cartesian monoidal structure on the arena H(S), which is the restriction of the symmetric monoidal structure on Spc(S).

2.4.9. Since the topos SpcNis(S) has universality of colimits, as does SpcA1 (S) by [Chap. 1, Proposition 2.3.4], we have: Proposition 2.4.10. The category H(S) has universality of colimits.

2.4.11. For each smooth S-scheme X, let MS(X) denote the motivic localization Lmot(hS(X)). [Chap. 1, Proposition 2.2.9] implies directly:

Proposition 2.4.12. The category H(S) is generated under sifted colimits by the spaces MS(X), where X is a smooth S-scheme which is affine (over Spec(Z)). 3. POINTED MOTIVIC SPACES 53

2.4.13. Suppose the scheme S is classical, and noetherian of finite Krull dimension (in the classical sense). In this case there is an ordinary category of motivic spaces constructed by Morel–Voevodsky [MV99], which can be viewed as the underlying ordinary category of an (∞, 1)-category (see [Rob14] or [Hoy15, Appendix C]). We have:

Proposition 2.4.14. If S is a classical noetherian scheme of finite Krull dimension, then the (∞, 1)-category H(S) coincides with the Morel–Voevodsky (∞, 1)-category of motivic spaces over S.

Proof. This follows directly from the observation that the site Sm/S is equivalent to the site of classical smooth S-schemes. Indeed, any smooth S-scheme X is flat over S, hence itself classical. 

3. Pointed motivic spaces

3.1. Pointed fibred spaces.

3.1.1. A pointed (fibred) space over a scheme S is a pointed object in the category Spc(S), i.e. a Spc pair (F, x), with F a fibred space over S, and x : eS → F a morphism from the terminal object.

Spc• Spc Spc Spc By definition it admits a zero object ptS := (eS , id), where id : eS → eS is the identity morphism.

We will write Spc(S)• for the arena of pointed spaces over S. By [Lur16, Ex. 4.8.1.20, Prop. 4.8.2.11], it has a canonical structure of Spc•-module arena, and is canonically equivalent to the base change Spc(S) ⊗Spc Spc•.

3.1.2. Consider the forgetful functor sending a pointed space (F, x) to its underlying space F. This admits a left adjoint, which freely adjoins a point to F; that is, it is given on objects by the assignment Spc F 7→ F+ := (F t eS , x) where x is the canonical point. • For a smooth S-scheme X, we write hS(X) := hS(X)+ for the pointed space represented by X.

3.1.3. It is clear that Spc(S)• is equivalent to the category of modules over the monad with Spc underlying endofunctor F 7→ F t eS . Since the latter commutes with contractible colimits, it follows that the forgetful functor (F, x) 7→ F is conservative6, and preserves and reflects contractible colimits. This monadic description also implies that every pointed space can be written as a colimit of a simplicial diagram with each term in the essential image of F 7→ F+:

Lemma 3.1.4. The category Spc(S)• is generated under sifted colimits by objects of the form F+, where F is a space over S.

6Recall that a functor is conservative if it reflects isomorphisms. 54 1. MOTIVIC SPACES AND SPECTRA

3.1.5. The cartesian monoidal structure on the arena Spc(S) induces a monoidal structure on Spc(S)• (see [Rob15, Cor. 2.32]):

Lemma 3.1.6. The arena Spc(S)• admits a canonical symmetric monoidal structure, which is uniquely characterized by the fact that the functor Spc(S) → Spc(S)• is symmetric monoidal. Further, we have the following universal property: Given any symmetric monoidal morphism of arenas u : Spc(S) → C, with C pointed, there exists a unique symmetric monoidal morphism of arenas u˜ : Spc(S)• → C, and an isomorphism u˜ ◦ (−)+ ≈ u.

Spc• Spc• We will write ⊗S for the monoidal product, and HomS for the internal hom. The Spc• Spc • monoidal unit is 1S := (eS )+ = hS(S).

3.1.7. Given a pointed space T over S, the T-suspension endofunctor ΣT := ΣT,S : Spc(S)• → Spc(S)• is defined by the assignment

(F, x) 7→ (F, x) ⊗S T.

Dually, the T-loop space endofunctor ΩT := ΩT,S is given by

(F, x) 7→ HomS(T, (F, x)).

These endofunctors form an adjunction (ΣT, ΩT).

3.2. Pointed motivic spaces.

3.2.1. A pointed space (F, x) over S is Nisnevich-local, A1-homotopy invariant, or motivic, if the underlying space F has the respective property. We write H(S)• for the full subcategory of Spc(S)• spanned by motivic pointed spaces. Note that this is equivalent to the category of pointed objects in H(S). • We write MS(X) := MS(X)+ for the pointed motivic space represented by a smooth S-scheme X.

3.2.2. The monadic description we have given of the category Spc(S)• also applies to H(S)•, so that in particular we have: Lemma 3.2.3. The forgetful functor (F, x) 7→ F, on the category of pointed motivic spaces over S, is conservative and preserves and reflects contractible colimits.

3.2.4. The symmetric monoidal structure on the arena Spc(S)• restricts to one on H(S)•, uniquely characterized by the fact that the morphism H(S) → H(S)• is symmetric monoidal. As in [Chap. 1, Lemma 3.1.6] we have the following universal property: Proposition 3.2.5. Given any symmetric monoidal morphism of arenas u : H(S) → C, with C pointed, there exists a unique symmetric monoidal morphism u˜ : H(S)• → C, and an isomorphism u˜ ◦ (−)+ ≈ u.

3.2.6. Note that the full subcategory H(S)• is a reflective localization of Spc(S)• at the small set of morphisms of the form • 1 • (3.1) hS(X × A ) → hS(X) for each smooth S-scheme X, and (3.2) lim h•(C(Xˇ /X)) → h•(X) −→ S α S n∈∆op for each smooth S-scheme X and Nisnevich covering family (fα :Xα → X)α. 4. MOTIVIC SPECTRA 55

Spc• Spc• In particular we obtain localization functors LNis := LNis,S,LA1 := LA1,S, and

Spc• (3.3) Lmot := Lmot,S : Spc(S)• → H(S)•. These admit descriptions which are completely analogous to their unpointed versions. According to the universal property in [Chap. 1, Lemma 3.1.6], these localization functors are symmetric monoidal morphisms of arenas, characterized by commutativity with the functor F 7→ F+. In particular, we have • • Lmot(hS(X)) = MS(X) for each smooth S-scheme X.

3.2.7. As a result of [Chap. 1, Proposition 2.4.12] and [Chap. 1, Lemma 3.1.4], we have:

Proposition 3.2.8. The category H(S)• is generated under sifted colimits by objects of the • form MS(X), for X a smooth S-scheme which is affine (over Spec(Z)).

4. Motivic spectra

4.1. Fibred spectra.

4.1.1. Let S be a scheme. Fix a pointed space T over S 7.A (fibred) T-spectrum over S is a T-spectrum object in the category Spc(S)• of pointed spaces over Sm/S.

That is, a T-spectrum is the data of a sequence (Fn)n>0 of fibred pointed spaces over S and structural isomorphisms ∼ αn : Fn −→ ΩT(Fn+1) for each integer n > 0, together with a homotopy coherent system of compatibilities between these isomorphisms.

4.1.2. We will write Spt T(S) for the arena of T-spectra over S, which is by definition the limit of the cofiltered diagram

ΩT ΩT (4.1) ··· −−→Spc(S)• −−→Spc(S)• in the category of (∞, 1)-categories and right adjoint functors. Equivalently, this is the colimit of the filtered diagram

ΣT ΣT (4.2) Spc(S)• −−→Spc(S)• −−→· · · in the category of arenas.

4.1.3. By construction, the adjunction (ΣT, ΩT) at the level of pointed spaces gives rise to an equivalence Spt Spt ΣT : Spt T(S)  Spt T(S) : ΩT , Spt where ΣT is given by the assignment (Fn)n 7→ (Fn+1)n (with the same structural isomorphisms), Spt and ΩT is given by (Fn)n 7→ (ΩT(Fn+1))n (with the induced structural isomorphisms). Spt Spt When there is no risk of ambiguity, we will write ΣT := ΣT and ΩT := ΩT .

7 1 1 1 For us T will be S , PS or S ⊗ Gm. 56 1. MOTIVIC SPACES AND SPECTRA

4.1.4. Also by construction, we have for each n > 0 a canonical functor ∞−n Ω : Spt T(S) → Spc(S)• ∞−n−1 ∞−n which is the projection to the nth term, with canonical isomorphisms ΩTΩT = ΩT . Dually we have a canonical functor ∞−n Σ : Spc(S)• → Spt T(S), ∞−n ∞−n−1 ∞ ∞−n left adjoint to ΩT , with canonical isomorphisms ΣT ΣT = ΣT . ∞ ∞−0 ∞ ∞−0 We will write ΩT := ΩT and ΣT := ΣT . Note that there are canonical isomorphisms ∞−n n ∞ ∞−n ∞ n ΣT = ΩTΣT and ΩT = ΩT ΣT for each n.

4.1.5. Note that for T = S1, the category of S1-spectra over S is equivalent to the category of 1 presheves of spectra on the site Sm/S; we will identify the two implicitly. (Here S denotes the constant pointed space over S valued in the 1-sphere (with its canonical base point).)

∞ ∞ • 4.1.6. For a smooth S-scheme X, we will write hT,S(X) := ΣT,S(hS(X)) for the infinite suspension spectrum of the pointed motivic space represented by X. This is the T-spectrum representing X, in the sense that for any T-spectrum E there are canonical functorial isomorphisms of pointed spaces (4.3) Maps (h∞ (X), E) ≈ Γ(X, Ω∞ (E)). Spt T (S) T,S T,S

4.1.7. By [Lur09b, Lem. 6.3.3.6], and the definition of Spt T(S) as the cofiltered limit of the diagram (4.1), we have:

Lemma 4.1.8. The category Spt T(S) is generated under filtered colimits by objects of the form ∞−n ΣT (F), for F a pointed space over S and n > 0.

4.1.9. Assume that for some n > 2, the pointed space T is n-symmetric, i.e. the cyclic permutation of T⊗n is homotopic to the identity morphism. In this case, the main result of [Rob15] endows the arena Spt T(S) with a canonical symmetric monoidal structure:

Lemma 4.1.10. The arena Spt T(S) admits a canonical symmetric monoidal structure, and the ∞ functor ΣT lifts to a symmetric monoidal morphism of arenas, which sends T to a monoidally invertible object of Spt T(S). Further, it satisfies the following universal property: given a symmetric monoidal arena D, and a symmetric monoidal morphism u : Spc(S)• → D sending T to a monoidally invertible object in D, there exists a unique symmetric monoidal morphism u˜ : Spt T(S) → D and an ∞ isomorphism u˜ ◦ ΣT = u.

Spt T Spt T We will write ⊗S := ⊗S for the monoidal product in Spt T(S), and HomS := HomS Spt T ∞ for the internal hom. The monoidal unit 1S := 1S is the T-spectrum hT,S(S).

∞ 4.1.11. We will abuse notation and write T also for the monoidally invertible object ΣT (T). Let ⊗0 Spt T ⊗(−1) ⊗(−n) ⊗(−1) ⊗n T = 1S and let T be a monoidal inverse to T; for n > 1, write T = (T ) . The universal property in [Chap. 1, Lemma 4.1.10] shows that there are canonical isomor- Spt ◦n ⊗n Spt ◦n ⊗−n phisms of functors (ΣT ) = T ⊗ (−) and (ΩT ) = T ⊗ (−) for each n > 0.

4.2. Motivic spectra. 4. MOTIVIC SPECTRA 57

4.2.1. We say that a T-spectrum E over S satisfies Nisnevich descent or A1-homotopy invariance ∞−n if for each n > 0, its nth component ΩT (E) satisfies the respective property (as a pointed fibred space). A motivic T-spectrum is a T-spectrum satisfying Nisnevich descent and A1-homotopy invariance.

4.2.2. Let SHT(S) denote the category of motivic T-spectra. This is equivalent to the category of Lmot(T)-spectra objects in the category of pointed motivic spaces over S. In particular it is an arena.

∞ ∞ 4.2.3. For a smooth S-scheme X, we will write MS (X) := MT,S(X) for the motivic T-spectrum ∞ • ∞−k ∞−k • ΣT (MS(X)), and similarly MT,S (X) := ΣT (MS(X)) for k > 0. By [Chap. 1, Proposition 2.4.12] and [Chap. 1, Lemma 4.1.8], we have:

Proposition 4.2.4. The category SHT(S) is generated under sifted colimits by objects of the −n ∞ form ΣT (MT,S(X)), for X a smooth S-scheme which is affine (over Spec(Z)) and n > 0.

4.2.5. Assuming that T is n-symmetric for some n > 2, the symmetric monoidal structure on the arena Spt T(S) ([Chap. 1, Lemma 4.1.10]) restricts to one on SHT(S).

4.2.6. Note that E satisfies Nisnevich descent (resp. A1-homotopy invariance) if and only if it is a local object with respect to the small set of morphisms

(4.4) lim Ωk h∞ (C(Xˇ /X) ) → Ωk h∞ (X) −→ T T,S α n T T,S n∈∆op for every smooth S-scheme X and integer k > 0 (resp. the small set of morphisms k ∞ 1 k ∞ (4.5) ΩT hT,S(X × A ) → ΩT hT,S(X) for every smooth S-scheme X, Nisnevich covering family (fα :Xα → X)α, and integer k > 0).

In other words, the full subcategory SHT(S) is an accessible localization of the arena Spt T(S), so that the inclusion admits a left adjoint

Spt T (4.6) Lmot := Lmot,S : Spt T(S) → SHT(S).

1 Spt T Spt T Similarly we have Nisnevich- and A -localization functors denoted LNis and LA1 , respectively.

4.2.7. These localization functors admit descriptions completely analogous to their counterparts at the level of motivic spaces. For example,

Spt T ∞ n (4.7) L 1 (E) = lim Hom (h (∆ ), E) A −→ S T,S S [n]∈∆op for every T-spectrum E. Using the universal property ([Chap. 1, Lemma 4.1.10]), they define symmetric monoidal morphisms of arenas which can be characterized uniquely by commutativity with the functor ∞ ΣT . In particular, we have ∞−k ∞−k Lmot(hT,S (X)) = MT,S (X) for each k > 0. 58 1. MOTIVIC SPACES AND SPECTRA

8 4.2.8. As soon as the category Spt T(S) is stable , we have a canonical isomorphism

(4.8) Lmot(E) = LA1 (LNis(E)) for any T-spectrum E. This follows from the fact that the condition of Nisnevich descent is defined by finite limits, and finite limits commute with colimits in stable (∞, 1)-categories.

8 1 1 1,× 1 This will be true in all the cases we consider, e.g. T = S , T = S ⊗ (MS(AS ), 1) = (MS(PS), ∞). 5. INVERSE AND DIRECT IMAGE FUNCTORIALITY 59

5. Inverse and direct image functoriality

5.1. For motivic spaces.

5.1.1. Let f :T → S be a morphism of schemes. The direct image functor

Spc (5.1) f∗ : Spc(T) → Spc(S) is defined as restriction along the base change functor Sm/S → Sm/T. ∗ Its left adjoint fSpc, the inverse image functor, is given by left Kan extension (see [Chap. 0, Proposition 2.3.7]). Hence it is uniquely characterized by commutativity with small colimits and the formula

∗ (5.2) fSpc(hS(X)) = hT(X × T) S for smooth S-schemes X.

5.1.2. Note that the base change functor Sm/S → Sm/T preserves Nisnevich covering families 1 ∗ and A -projections. It follows that the inverse image functor fSpc preserves Nisnevich-local equivalences and A1-homotopy equivalences.

Spc 1 By adjunction, its right adjoint f∗ preserves Nisnevich-local and A -homotopy invariant H H spaces and induces a functor f∗ : H(T) → H(S). This admits a left adjoint f∗ given by the formula ∗ ∗ fH(F) = Lmot(fSpc(F)).

5.1.3. Both the direct and inverse image functors are symmetric monoidal:

Spc H Lemma 5.1.4. The functor f∗ (resp. f∗ ) admits a canonical symmetric monoidal structure.

Proof. Since the respective symmetric monoidal structures are cartesian, it suffices to show that f∗ commutes with finite products. In fact, it commutes with arbitrary small limits since it is a right adjoint. 

∗ ∗ Lemma 5.1.5. The functor fSpc (resp. fH) admits a canonical symmetric monoidal structure.

Proof. By adjunction from [Chap. 1, Lemma 5.1.4], we obtain a canonical structure of colax symmetric monoidal functor on f ∗. That is, there are canonical morphisms

(5.3) f ∗(F × G) → f ∗(F) × f ∗(G) S T for any two spaces F and G over S. It suffices to show that these morphisms are invertible. ∗ Since fSpc commutes with small colimits, and the cartesian product commutes with small colimits in each argument, one reduces to the case of representables, in which case the claim is ∗ clear. For fH, the claim follows from the first because motivic localization commutes with finite products ([Chap. 1, Proposition 2.4.7]). 

5.2. For pointed motivic spaces. 60 1. MOTIVIC SPACES AND SPECTRA

Spc 5.2.1. Let f :T → S be a morphism of schemes. Since f∗ preserves the terminal object, it Spc• induces a functor f∗ : Spc(T) → Spc(S) given on objects by the formula

Spc• Spc Spc f∗ (G, y) = (f∗ (G), f∗ (y)).

Its left adjoint f ∗ : Spc(T) → Spc(S) is uniquely characterized, according to [Chap. 1, Spc• Lemma 3.1.4], by the fact that it commutes with sifted colimits and with the functor F 7→ F+: (5.4) f ∗ (F ) = f ∗ (F) Spc• + Spc + for any space F over S. Explicitly, it is given on objects by the formula f ∗ (F, x) = (f ∗ (F), f ∗ (x)) Spc• Spc Spc for each pointed space (F, x) over S. Lemma 5.2.2. The inverse image functor f ∗ admits a canonical symmetric monoidal Spc• structure.

Proof. This follows directly from the universal property [Chap. 1, Lemma 3.1.6] and the formula (5.4). 

Spc• 1 5.2.3. The direct image functor f∗ preserves the properties of Nisnevich descent and A - homotopy invariance, and induces a functor f H. Its left adjoint f ∗ is given by composing f ∗ ∗ H• Spc• with the motivic localization functor: (5.5) f ∗ := L f ∗ . H• mot Spc•

5.3. For motivic spectra. In this paragraph we fix, for each scheme S, a pointed space TS ∗ over S, together with isomorphisms f (TS) → TT for any morphism f :T → S, and a Spc• homotopy coherent system of compatibilities between these isomorphisms. Such a datum defines an object of the limit lim Spc(S) over all schemes S. Of course, this ←−S • limit is equivalent to the category Spc(Spec(Z))•, so that choosing such a datum is equivalent to choosing a pointed space over Spec(Z) and defining TS to be its inverse image over each scheme S. ∞ ∞ For simplicity, we will drop TS from the subscripts in the notation Spt(S), SH(S), Σ ,Ω , etc., when there is no risk of confusion.

5.3.1. Let f :T → S be a morphism of schemes. Since f ∗ is monoidal ([Chap. 1, Lemma 5.2.2]), Spc• it commutes with T-suspensions. ∗ We let fSpt denote the unique morphism of arenas making the diagram

ΣT ΣT Spc(S)• Spc(S)• ··· Spt(S)

f ∗ f ∗ f ∗ Spc• Spc• Spt

ΣT ΣT Spc(T)• Spc(T)• ··· Spt(T)

∞ commute. That is, it is the unique morphism of arenas which commutes with ΣT . Using the universal property stated in [Chap. 1, Lemma 4.1.10], we get:

∗ Lemma 5.3.2. The functor fSpt admits a canonical symmetric monoidal structure. 6. SMOOTH MORPHISMS 61

∗ 5.3.3. Alternatively, we can use [Chap. 1, Lemma 4.1.8] to describe fSpt as the unique functor −n ∞ which commutes with filtered colimits and with the functor ΣT ΣT for each n > 0: ∗ −n ∞ −n ∞ ∗ (5.6) fSpt (ΣT ΣT (F)) = ΣT ΣT (fSpt (F)) for each pointed space F over S.

Spt ∗ 5.3.4. Let f∗ be the right adjoint of fSpt . This can be described as the unique limit-preserving functor which makes the diagram

ΩT ΩT Spt(T) ··· Spc(T)• Spc(T)•

Spt Spc• Spc• f∗ f∗ f∗

ΩT ΩT Spt(S) ··· Spc(S)• Spc(S)• commute, i.e. which commutes with Ω∞:

∞ Spt Spc• ∞ (5.7) Ω f∗ = f∗ Ω and is given on objects by the assignment

Spc• E = (Fn)n 7→ f∗(E) = (f∗ (Fn))n.

Spt SH 5.3.5. The direct image functor f∗ preserves motivic spectra and induces a functor f∗ . ∗ ∗ We let fSH be its left adjoint, the symmetric monoidal functor LmotfSpt . This is the unique ∞ symmetric monoidal morphism of arenas which commutes with ΣT : H(S)• → SH(S).

6. Smooth morphisms

6.1. The functor p].

6.1.1. Let p :X → S be a smooth morphism. In this case the base change functor admits a right adjoint, the forgetful functor Sm/X → Sm/S: p (Y → X) 7→ (Y → X −→ S).

∗ It follows that the functor pSpc coincides with restriction along the forgetful functor, and admits a left adjoint Spc p] : Spc(T) → Spc(S) which is defined by left Kan extension (see [Chap. 0, Proposition 2.3.7]), and hence is uniquely characterized by commutativity with small colimits and the formula Spc (6.1) p] (hX(Y)) = hS(Y). for smooth X-schemes Y.

6.1.2. Since the forgetful functor Sm/X → Sm/S preserves Nisnevich covering families and 1 Spc 1 A -projections, it follows that p] preserves Nisnevich-local equivalences and A -homotopy equivalences. ∗ 1 In particular its right adjoint pSpc preserves Nisnevich descent and A -homotopy invariance, ∗ and induces a morphism pH on motivic spaces. H Spc Its left adjoint p] is given by applying p] and then the localization functor Lmot: H Spc p] (F) = Lmot(p] (F)). 62 1. MOTIVIC SPACES AND SPECTRA

∗ 6.1.3. Let p :X ,→ S be a smooth morphism. By [Chap. 1, Lemma 5.1.5], the functors pSpc and ∗ Spc pH admit canonical symmetric monoidal structures, so that their respective left adjoints p] H and p] admit colax symmetric monoidal structures. If p is an open immersion, then these monoidal structures are strict: Proposition 6.1.4. Let j :U ,→ X be a quasi-compact open immersion. Then the canonical Spc H colax symmetric monoidal structure on the functor j] (resp. j] ) is strict.

Proof. It suffices to show that the canonical morphisms

Spc Spc Spc j] (F × G) → j] (F) × j] (G) U S Spc are invertible for all spaces F and G on U. Since j] commutes with small colimits, and the cartesian product commutes with small colimits in each argument, one reduces to the case of representables.

Then the claim follows from the fact that the fibred products X ×U Y and X ×S Y are canonically identified (since j is a monomorphism, i.e. its diagonal morphism is invertible). H As above, the claim for j] follows the fact that motivic localization commutes with finite products. 

6.1.5. Let (fα :Sα → S)α be a Nisnevich covering family. Given a morphism of motivic spaces over S, the following proposition says that it is invertible if and only if its inverse image on each Sα is invertible: Proposition 6.1.6 (Nisnevich separation). Let S be a scheme. For any Nisnevich covering ∗ family (pα :Sα → S)α, the family of inverse image functors (pα)H : H(S) → H(Sα) is conservative.

This is in fact true at the level of Nisnevich-local spaces, which is what we will prove.

Proof. Let ϕ : F1 → F2 be a morphism of Nisnevich-local spaces on S, and suppose that the following condition holds: ∗ ∗ (∗) For each α, the morphism (pα)Nis(F1) → (pα)Nis(F2) is invertible. The claim is that under this assumption, the morphism

Γ(X, F1) → Γ(X, F2) is invertible for every smooth S-scheme X.

Since Fi satisfy Nisnevich descent, it suffices to show that the morphism

(6.2) Γ(Xα, F1) → Γ(Xα, F2) is invertible for each α, where Xα is the base change of X along pα. ∗ Since hS(Xα) = (pα)](pα) (hS(X)), we have by adjunction ∗ Γ(Xα, Fi) = Γ(X, (pα)](pα) Fi) for each α and i.

Hence the claim follows from the assumption (∗). 

6.2. Smooth base change formulas. 6. SMOOTH MORPHISMS 63

6.2.1. Suppose we have a cartesian square

f 0 T0 S0

p0 p f T S of schemes. At the level of (motivic) spaces, pointed spaces, and spectra, there are canonical 2-morphisms 0 0 ∗ ∗ (6.3) (p )](f ) → f p], ∗ 0 ∗ 0 (6.4) f∗p → (p ) (f )∗, constructed in [Chap. 2, Paragraph 2.2]. The following says that Spc and H satisfy the left base change property along smooth morphisms (see loc. cit.): Proposition 6.2.2. If p and p0 are smooth, then the 2-morphisms (6.3) and (6.4) are invertible at the level of spaces and motivic spaces.

Proof. It suffices to consider the first 2-morphism; the second is its right transpose. For fibred spaces, we note that the functors in question commute with small colimits, so that we may reduce to representable spaces, in which case the claim is obvious. Similarly, for motivic spaces we use [Chap. 1, Proposition 2.4.12] to reduce to the case of motivic localizations of representable spaces. 

6.2.3. Next we consider the case of pointed spaces. Then we have: Proposition 6.2.4. If p and p0 are smooth, then the 2-morphisms (6.3) and (6.4) are invertible at the level of pointed spaces and pointed motivic spaces.

0 0 ∗ ∗ Proof. By transposition it suffices to show that the canonical morphism (p )](f ) → f p] is invertible. Since the functors in question commute with colimits and with the functor F 7→ F+, the claim follows from [Chap. 1, Lemma 3.1.4] (resp. [Chap. 1, Proposition 3.2.8]) and smooth base change for unpointed spaces ([Chap. 1, Proposition 6.2.2]). 

6.2.5. Fix a family of pointed fibred spaces (TS)S as in [Chap. 1, Paragraph 5.3]. We have: Proposition 6.2.6. If p and p0 are smooth, then the 2-morphisms (6.3) and (6.4) are invertible at the level of T-spectra and motivic T-spectra.

Proof. This follows from [Chap. 1, Lemma 4.1.8] (resp. [Chap. 1, Proposition 4.2.4]) and smooth base change for pointed spaces ([Chap. 1, Proposition 6.2.4]). 

6.3. Smooth projection formulas. Let p :X → S be a smooth morphism. Note that the ∗ symmetric monoidal functor pSpc endows Spc(X) with a structure of Spc(S)-module category. The following verifies the left projection formula along smooth morphisms, in the sense of [Chap. 2, Paragraph 2.2]:

Spc H Proposition 6.3.1. The functor p] (resp. p] ) lifts to a morphism of Spc(S)-module categories (resp. H(S)-module categories). In other words, there are canonical isomorphisms ∗ (6.5) p](G × p (F)) → p](G) × F X S 64 1. MOTIVIC SPACES AND SPECTRA and dually ∗ (6.6) HomS(p](G), F) → p]HomX(G, p (F)) for any fibred spaces (resp. motivic spaces) F over S and G over X.

We recall how to use the monoidal structure on p∗ to construct the morphism (6.5):

Spc ∗ The counit of the adjunction (p] , pSpc) induces a canonical morphism ∗ ∗ ∗ ∼ ∗ G × p (F) → p p](G) × p (F) −→ p (p](G) × F) X X S which corresponds by adjunction to the morphism desired.

Proof. It suffices to show that the canonical morphism (6.5) is invertible. For fibred spaces, we may reduce to the case where the spaces F and G are representable, since the functions involved commute with small colimits. In this case the claim is clear. The case of motivic spaces is similar, using [Chap. 1, Proposition 2.4.12] to reduce to the case of motivic localizations of representable spaces. 

6.3.2. The following slightly more general formula, proved in exactly the same way, will also be useful: Lemma 6.3.3. Let p :X → S be a smooth morphism. Let G be a fibred space (resp. motivic space) over X, and F → F0 a morphism of fibred spaces (resp. motivic spaces) over S. Then there is a canonical isomorphism ∗ ∼ (6.7) p](G × p (F)) −→ p](G) × F p∗(F0) F0 of fibred spaces (resp. motivic spaces) over S.

6.3.4. Similarly we get smooth projection formulas for pointed spaces and spectra. As above, the following statements are equivalent to formulas of the form (6.5) and (6.6).

Spc• H• Proposition 6.3.5. The functor p] (resp. p] ) lifts to a morphism of Spc(S)•-module categories (resp. H(S)•-module categories).

Proof. This follows from [Chap. 1, Lemma 3.1.4] (resp. [Chap. 1, Proposition 3.2.8]) and the smooth projection formula for unpointed spaces ([Chap. 1, Proposition 6.3.1]). 

Fix a family of pointed fibred spaces (TS)S as in [Chap. 1, Paragraph 5.3]. Then we have: Spt SH Proposition 6.3.6. The functor p] (resp. p] ) lifts to a morphism of Spt(S)-module categories (resp. SH(S)-module categories).

Proof. This follows from [Chap. 1, Lemma 4.1.8] (resp. [Chap. 1, Proposition 4.2.4]) and the smooth projection formula for pointed spaces ([Chap. 1, Proposition 6.3.5]). 

7. Closed immersions

7.1. A topological digression, I: local cocontinuity. In order to prove [Chap. 1, Propo- sition 7.3.2], the main result of this section, we will begin by making a small topological digression. Recall that a functor u between sites is topologically cocontinuous9 if the restriction of presheaves functor u∗ preserves local equivalences. In this paragraph we introduce a slightly

9The term cocontinuous is used in [SGA 4]. 7. CLOSED IMMERSIONS 65 weaker version of this condition, where the functor u is only “locally cocontinuous” with respect to a weaker topology. We will show that for such functors, the restriction functor u∗ preserves local equivalences between sheaves for the weaker topology.

7.1.1. Let C be a small (∞, 1)-category. As in [Chap. 0, Sect. 2], we will write P(C) for the (∞, 1)-category of presheaves on C, and h : C ,→ P(C) for the Yoneda embedding. Given a topology τ on C, we will write Shτ (C) for the subcategory of τ-sheaves, i.e. the presheaves F for which the canonical morphism F(c) → Maps(R, F) is invertible for all τ-covering sieves R ,→ h(c) of all objects c ∈ C. We will write incτ : Shτ (C) ,→ P(C) for the inclusion, and Lτ : P(C) → Shτ (C) for the left-exact left adjoint (the τ-localization functor). Given a functor u : C → D, we will write u∗ : P(D) → P(C) for the restriction of presheaves functor, and u! (resp. u∗) for the left adjoint (resp. right adjoint).

7.1.2. Let (C, τ) and (D, τ 0) be sites. Recall that a functor u : C → D is topologically cocontinuous if the following condition is satisfied: (COC) For every τ 0-covering sieve R0 ,→ h(u(c)), the sieve R ,→ h(c), generated by morphisms c0 → c such that h(u(c0)) → h(u(c)) factors through R0, is τ-covering. Note that R can be described as the sieve (7.1) R = u∗(R0) × h(c) ,→ h(c) u∗(h(u(c))) obtained from R0 ,→ h(u(c)) by applying u∗ and taking the base change along the unit morphism ∗ ∗ h(c) → u u!(h(c)) = u h(u(c)).

0 0 0 7.1.3. Let τ0 be a topology on D which is weaker than τ . For simplicity we will assume that τ0 0 is subcanonical, so that representable presheaves are τ -sheaves. Let L 0 denote the associated 0 τ0 0 τ0-sheaf functor, left adjoint to the inclusion. 0 We will say that u is τ0-locally topologically cocontinuous if it satisfies the following weaker version of the condition (COC): (COC’) For every τ 0-covering sieve R0 ,→ h(u(c)), the sieve R ,→ h(c), generated by 0 0 0 0 morphisms c → c such that h(u(c )) → h(u(c)) factors through the τ0-sheaf associated to R , is τ-covering. Note that R can be described as the sieve ∗ 0 (7.2) R := u (Lτ 0 (R )) × h(c) ,→ h(c) 0 u∗(h(u(c)))

0 obtained in the same way as (7.1) starting from L 0 (R ) ,→ h(u(c)). τ0

7.1.4. The following lemma is proved in exactly the same way as the analoguous result for topologically cocontinuous functors [SGA 4, Exp. III, Prop. 2.2]:

0 0 Lemma 7.1.5. Let u :(C, τ) → (D, τ ) be a functor. Let τ0 be a topology on D which is weaker 0 0 than τ . If u is τ0-locally cocontinuous, then the functor

∗ ∗ u u : Sh 0 (D) ,→ P(D) −→P(C) 0 τ0 sends τ 0-local equivalences to τ-local equivalences. 66 1. MOTIVIC SPACES AND SPECTRA

∗ Proof. First of all, note that u0 admits a left adjoint

L 0 0 u! τ0 u : P(C) −→P(D) −−→Sh 0 (D) ! τ0 by construction. Let R0 ,→ h(d) be a τ 0-covering sieve of an object d of D. To show that ∗ 0 ∗ ϑ : u0(R ) ,→ u0(h(d)) is a τ-local equivalence, it suffices by universality of colimits to show ∗ that, for every object c of C and every morphism ϕ : h(c) → u0h(d), the base change u∗R0 × h(c) ,→ h(c) 0 ∗ u0 h(d) is a τ-covering sieve. Note that ϕ factors canonically through the unit morphism h(c) → ∗ 0 ∗ ∗ 0 ∗ ∗ u0u! h(c) = u0h(u(c)) and the canonical morphism u0u! h(c) = u0h(u(c)) → u0h(d) (obtained ∗ ∗ ∗ by adjunction from ϕ). The base change of ϑ by u0h(u(c)) → u0h(d) is identified, since u0 commutes with limits, with the canonical morphism

∗ 0 ∗ u (R × h(u(c))) → u0h(u(c)). h(d)

0 0 0 Since R ×h(d) h(u(c)) ,→ h(u(c)) is τ -covering, as the base change of a τ -covering sieve, the conclusion follows by applying the condition (COC’). 

7.2. A topological digression, II: contractible colimits. In this paragraph we consider a topology whose associated category of sheaves coincides with the free completion by contractible colimits. This topology, which we denote cdm, is the one associated to the minimal cd-structure10, with no commutative squares. We show that for any cdm-locally topologically cocontinuous functor u, the restriction functor u∗ on sheaves commutes with contractible colimits.

7.2.1. Given an (∞, 1)-category C admitting an initial object ∅C, the cdm topology is defined by op the sieve ∅ ,→ h(∅C), where ∅ is the initial presheaf. A presheaf F : C → Spc is a cdm-sheaf if and only if the space F(∅C) is contractible. Let Shcdm(C) denote the (∞, 1)-category of cdm-sheaves. It is not difficult to show that Shcdm(C) is the (∞, 1)-category freely generated by C under contractible colimits.

Let Lcdm denote the cdm-localization functor, left adjoint to the inclusion. For a presheaf F on C,Lcdm(F) can be described as the unique cdm-sheaf for which the space Lcdm(F)(c) is identified with F(c) whenever c is not initial.

7.2.2. For convenience we state an easy-to-use sufficient condition for cdm-local cocontinuity.

Lemma 7.2.3. Let (C, τ) and (D, τ 0) be ∞-sites and u : C → D a functor. Assume that D 0 admits an initial object ∅D, and that the topology τ is stronger than cdm. Then for the functor u to be cdm-locally cocontinuous, the following condition is sufficient: 0 0 (COC’cdm) For every τ -covering sieve R ,→ h(u(c)), the sieve R ,→ h(c), generated by morphisms c0 → c such that either h(u(c0)) → h(u(c)) factors through R0 ,→ h(u(c)) or u(c0) is initial, is τ-covering.

Indeed let c0 → c be a morphism such that u(c0) is initial. Then the (unique) morphism 0 h(u(c )) = Lcdm(∅) → h(u(c)) factors as the composite of the (unique) morphism Lcdm(∅) → 0 0 Lcdm(R ) and Lcdm(R ) → h(u(c)).

10See [Voe10] for the notion of cd-structure. 7. CLOSED IMMERSIONS 67

7.2.4. The following lemma is a formal consequence of [Chap. 1, Lemma 7.1.5], and the fact that cdm-sheaves are stable by contractible colimits: Lemma 7.2.5. Suppose that u is cdm-locally cocontinuous. Then the functor

∗ incτ0 u Lτ Shτ 0 (D) ,−→ P(D) −→P(C) −−→Shτ (C) commutes with contractible colimits.

0 Proof. Since the topology τ is a refinement of the cdm topology, the inclusion incτ 0 factors as

cdm cdm incτ0 inc incτ 0 : Shτ 0 (D) ,−→ Shcdm(D) ,−→ P(D).

Similarly the left adjoint Lτ 0 factors as

cdm cdm L Lτ0 L 0 : P(D) −−−→Sh 0 (D) −−−→Sh 0 (D), τ τcdm τ cdm cdm and Lτ 0 is left adjoint to incτ 0 . 0 Given a diagram (Fi)i∈I of τ -sheaves indexed by a contractible (∞, 1)-category I, consider the counit morphism cdm cdm cdm cdm lim inc 0 (F ) −→ inc 0 L 0 lim inc 0 (F ), −→ τ i τ τ −→ τ i i i 0 ∗ ∗ which is clearly a τ -local equivalence. By applying ucdm = u inccdm this induces a morphism ∗ cdm ∗ cdm cdm cdm u lim inc 0 (F ) −→ u inc 0 L 0 lim inc 0 (F ), cdm −→ τ i cdm τ τ −→ τ i i i which is identified with a canonical morphism ∗ ∗ lim u inc 0 (F ) −→ u inc lim F −→ τ i τ −→ i i i since the inclusion inccdm commutes with contractible colimits. By [Chap. 1, Lemma 7.1.5], this is a τ-local equivalence, so the claim follows. 

7.3. The exceptional inverse image functor i!.

7.3.1. Let i :Z ,→ S be a closed immersion. Note that if the base change functor Sm/S → Sm/Z were topologically cocontinuous (see [Chap. 1, Paragraph 7.1]), then the direct image functor i∗ on Nisnevich sheaves would commute with arbitrary small colimits. Though this is not quite true, we will show that this is true cdm-locally (see [Chap. 1, Paragraph 7.2]), which will imply that i∗ commutes with contractible colimits: H Proposition 7.3.2. Let i :Z ,→ S be a closed immersion. Then the direct image functor i∗ commutes with contractible colimits.

Proof. By [Chap. 1, Lemma 7.2.5] it suffices to show that the base change functor Sm/S → Sm/Z is cdm-locally cocontinuous. For this it suffices to check the condition (COC’cdm) of [Chap. 1, Lemma 7.2.3], which amounts to the following: 0 (∗) For any smooth S-scheme X and any Nisnevich covering sieve R of XZ, the sieve R of X 0 0 generated by morphisms X → X such that either (i) the empty sieve on XZ is Nisnevich-covering, 0 0 or (ii) XZ → XZ factors through R , is Nisnevich-covering. This condition follows directly from [Chap. 0, Proposition 5.6.2], which says that ´etale morphisms can be lifted (Zariski-locally) along i. 

In particular:

H• SH Corollary 7.3.3. The direct image functor i∗ (resp. i∗ ) commutes with small colimits. 68 1. MOTIVIC SPACES AND SPECTRA

By the adjoint functor theorem (see [Chap. 0, Paragraph 2.6.11]) we have a right adjoint i! (resp. i! ), called the exceptional inverse image functor. H• SH

7.4. The localization theorem. In this paragraph, we will work in the category of motivic spaces, and will omit the decoration H from the notation for simplicity.

7.4.1. Let i :Z ,→ S be a closed immersion with quasi-compact open complement j :U ,→ S. In this situation, we can construct a canonical commutative square

∗ j]j (F) F (7.3)

∗ j](eU) i∗i (F). of motivic spaces over S; see [Chap. 2, Paragraph 3.3]. We call this the localization square associated to the pair (i, j).

7.4.2. The main theorem is this chapter is the following, due to [MV99] over classical bases (noetherian and of finite dimension). Theorem 7.4.3. Let i :Z ,→ S be a closed immersion with quasi-compact open complement j :U ,→ S. Then for every motivic space F over S, the localization square (3.2) is cocartesian.

The proof will occupy [Chap. 1, Sect. 9].

7.4.4. We can deduce from [Chap. 1, Theorem 7.4.3] a pointed version: Corollary 7.4.5 (Localization). Let i :Z ,→ S be a closed immersion, with quasi-compact open complement j :U ,→ S. For any pointed motivic space (F, s) over S, there is a canonical cofibre sequence ∗ ∗ (7.4) j]j (F, s) → (F, s) → i∗i (F, s). and dually, a canonical fibre sequence ! ∗ (7.5) i∗i (F, s) → (F, s) → j∗j (F, s) of motivic spaces over S.

Proof. We want to show that the commutative square of pointed motivic spaces

∗ j]j (F, x) (F, x)

Spc• ∗ ptS i∗i (F, x) is cocartesian. Since the forgetful functor (F, x) 7→ F reflects contractible colimits ([Chap. 1, Lemma 3.2.3]), it suffices to show that the induced square of underlying motivic spaces

∗ ∗ j]j F tj]j (eS) eS F

H ∗ eS i∗i F. is cocartesian. 7. CLOSED IMMERSIONS 69

Consider the composite square

∗ ∗ ∗ j]j F (j]j F) tj]j (eS) eS F

∗ ∗ j]j (eS) eS i∗i F. which is cocartesian by [Chap. 1, Theorem 7.4.3]. Since the left-hand square is evidently cocartesian, it follows that the right-hand square is also cocartesian. 

7.4.6. Similarly we also deduce localization for motivic spectra: Corollary 7.4.7. Let i :Z ,→ S be a closed immersion, with quasi-compact open complement j :U ,→ S. For any motivic T-spectrum E over S, there is a canonical cofibre sequence ∗ ∗ (7.6) j]j (E) → E → i∗i (E), and dually a fibre sequence ! ∗ (7.7) i∗i (E) → E → j∗j (E), of motivic T-spectra over S.

Proof. It suffices to show the first sequence is a cofibre sequence. Since the functors in question commute with small colimits, [Chap. 1, Proposition 4.2.4] allows us to the reduce to the case of pointed motivic spaces, which is [Chap. 1, Corollary 7.4.5]. 

7.4.8. An immediate corollary of [Chap. 1, Theorem 7.4.3] is: Corollary 7.4.9. Let i :Z ,→ S be a closed immersion with quasi-compact open complement. H H• SH Then the direct image functor i∗ (resp. i∗ , i∗ ) is fully faithful.

H• SH H Proof. The claims for i∗ and i∗ follow directly from that of i∗ . ∗ Considering the localization square for i∗(F), we see that the canonical morphism i∗i i∗ → i∗ is invertible. Hence it suffices to show that i∗ is conservative.

For this, let ϕ : F1 → F2 be a morphism of motivic spaces over Z such that i∗(ϕ) is invertible. To show that ϕ is invertible, it suffices to show that

Γ(X, F1) → Γ(X, F2) is invertible for each smooth Z-scheme X. By [Chap. 0, Proposition 5.6.2], we may assume that X is the base change of a smooth S-scheme Y. In this case the claim follows by assumption, since Γ(X, Fi) = Γ(Y, i∗(Fi)) for each i, by adjunction. 

7.5. Closed base change formula.

7.5.1. Let Θ be a cartesian square

k XZ X (7.8) g f Z i S, of schemes, with i and k closed immersions with quasi-compact open complements. 70 1. MOTIVIC SPACES AND SPECTRA

At the level of motivic spaces, there is a canonical 2-morphism ∗ ∗ (7.9) k∗g → f i∗ constructed in [Chap. 2, Paragraph 2.3]. The following says that H satisfies the right base change property along closed immersions (see loc. cit.): Corollary 7.5.2. The 2-morphism (7.9) is invertible at the level of motivic spaces.

Proof. This follows by considering the localization squares associated to the closed immersions j and k, respectively, and using the smooth base change formula ([Chap. 1, Proposition 6.2.2]). 

! 7.5.3. In the pointed setting, the functor i∗ admits a right adjoint i ([Chap. 1, Corollary 7.3.3]), so we obtain another 2-morphism by right transposition from (7.9). Hence we have: Corollary 7.5.4. Given a cartesian square of the form (7.8), the canonical 2-morphisms ∗ ∗ (7.10) k∗g → f i∗ ! ! (7.11) i f∗ → g∗k are invertible at the level of pointed motivic spaces.

7.5.5. Fixing a family of pointed fibred spaces (TS)S as in [Chap. 1, Paragraph 5.3], we have: Corollary 7.5.6. Given a cartesian square of the form (7.8), the canonical 2-morphisms ∗ ∗ (7.12) k∗g → f i∗ ! ! (7.13) i f∗ → g∗k are invertible at the level of motivic spectra.

7.6. Closed projection formula. Let i :Z ,→ S be a closed immersion. Note that the ∗ symmetric monoidal functor iSpc endows H(Z) with a structure of H(S)-module category. The following verifies the right projection formula along closed immersions, in the sense of [Chap. 2, Paragraph 2.3]: H Proposition 7.6.1. The functor i∗ lifts to a morphism of H(S)-module categories. In other words, there are canonical isomorphisms ∗ (7.14) i∗(G × i (F)) → i∗(G) × F Z S for any motivic spaces F over S and G over Z, and dually ! ∗ ! (7.15) i HomS(G, F) → HomZ(i G, i F) for any motivic spaces F and G over S.

Proof. The second isomorphism is the right transpose of the first. The first follows from the localization theorem ([Chap. 1, Corollary 7.4.5]) and the smooth projection formula. 

7.6.2. Similarly we get closed projection formulas for pointed motivic spaces and spectra. As above, the following statements are equivalent to formulas of the form (7.14) and (7.15). The proofs are completely analogous to those of [Chap. 1, Proposition 6.3.5] and [Chap. 1, Proposition 6.3.6].

H• Corollary 7.6.3. The functor i∗ lifts to a morphism of H(S)•-module categories.

Fix a family of pointed fibred spaces (TS)S as in [Chap. 1, Paragraph 5.3]. Then we have: SH Corollary 7.6.4. The functor i∗ lifts to a morphism of SH(S)-module categories. 7. CLOSED IMMERSIONS 71

7.7. Smooth-closed base change formula.

7.7.1. Let Θ be a cartesian square of schemes k XZ X (7.16) q p Z i S, where i and j are closed immersions with quasi-compact open complements, and p and q are smooth. There are canonical 2-morphisms

(7.17) p]k∗ → i∗q] (7.18) i!q∗ → k!q∗ at the level of pointed motivic spaces, constructed in [Chap. 2, Paragraph 2.4]. The following verifies the bidirectional base change property with respect to smooth mor- phisms and closed immersions: Corollary 7.7.2 (Smooth-closed base change). Given a cartesian square of the form (7.16), the 2-morphisms (7.17) and (7.18) are invertible at the level of pointed motivic spaces.

Proof. The second transformation is obtained by passing to right adjoints from the first. For the first, it suffices by [Chap. 1, Corollary 7.4.9] it suffices to show that the transformation ∗ ∗ p]k∗k → i∗q]k , obtained by pre-composition with k∗, is invertible. This follows directly from [Chap. 1, Corollary 7.4.5] and smooth base change.  72 1. MOTIVIC SPACES AND SPECTRA

8. Thom spaces

8.1. Thom spaces.

8.1.1. Let E be a vector bundle over a scheme S. Denote by p :E → S the projection and by s :S ,→ E the zero section.

We denote by thomE/S the Thom suspension endofunctor on the category of pointed motivic spaces, defined by the assignment

H• H• F 7→ thomE/S(F) := p] s∗ (F). Its right adjoint thomE/S, the Thom desuspension endofunctor, is given by F 7→ thomE/S(F) := s! p∗ (F). H• H• The Thom space of E is the pointed motivic space

E/S H• ThS(E) := thom (1S ).

8.1.2. We recall some results about Thom spaces that follow immediately from the various base change and projection formulas available to us. See [Chap. 2, Paragraph 3.4], where we will provide axiomatic proofs of these statements. Lemma 8.1.3. For each pointed motivic space F over S, there are canonical isomorphisms

(8.1) thomE/S(F) = F ⊗S ThS(E), E/S (8.2) thom (F) = HomS(ThS(E), F). Lemma 8.1.4. For every vector bundle E over S, there are canonical isomorphisms ∗ ∼ ∗ f ◦ thomE/S −→ thomE ×S T/T ◦ f ,

E ×S T/T ∼ E/S f∗ ◦ thom −→ thom ◦ f∗. In particular, for each pointed motivic space F over S, there are canonical isomorphisms ∗ ∼ f (ThS(E)) −→ ThT(E × T). S Lemma 8.1.5. Let p :X → S be a smooth morphism. For each vector bundle E over S, there are canonical isomorphisms

(8.3) p] ◦ thom(E ×S X)/X = thomE/S ◦ p], (8.4) thomE ×S X/X ◦ p∗ = p∗ ◦ thomE/S. Lemma 8.1.6. Let i :Z ,→ S be a closed immersion. For any vector bundle E over S, there are canonical isomorphisms

(8.5) i∗ ◦ thom(E ×S Z)/Z = thomE/S ◦ i∗, (8.6) i! ◦ thomE/S = thom(E ×S Z)/Z ◦ i!. Lemma 8.1.7. Let E0 → E → E00 be an exact sequence of vector bundles over S. Then there is a canonical 2-isomorphism ∼ thomE/S −→ thomE00/S ◦ thomE0/S. In particular, there is a canonical isomorphism of pointed motivic spaces ∼ 00 0 (8.7) ThS(E) −→ ThS(E ) ⊗S ThS(E ).

8.2. The Thom space of A1. The localization theorem allows us to give a more explicit 1 description of the Thom space, which we will allow us to compute ThS(AS) as the motivic space 1 represented by PS. 8. THOM SPACES 73

8.2.1. Lemma 8.2.2. For any vector bundle E on S, there is a canonical isomorphism of pointed motivic spaces11 × ∼ Cofib(MS(E ) → MS(E)) −→ ThS(E), where E× denotes the open subscheme complementary to the zero section.

× Proof. Let j :E ,→ E denote the open immersion. Applying the cocontinuous functor p] to the localization cofibre sequence associated to the closed immersion s, and evaluating the result on the motivic space 1E, we get the cofibre sequence • × • MS(E ) → MS(E) → ThS(E) of pointed motivic spaces over S. On underlying motivic spaces, this induces a cocartesian square × MS(E ) MS(E)

× MS(E ) t eS MS(E) t eS

eS ThS(E) as claimed. 

1 8.2.3. Let ΣS1 denote the S -suspension functor. Corollary 8.2.4. There are canonical isomorphisms 1 ∼ 1,× ∼ 1 (8.8) (MS(PS), ∞) ←− ΣS1 (MS(AS )) −→ ThS(AS) of pointed motivic spaces over S.

1 Proof. The right-hand isomorphism follows from [Chap. 1, Lemma 8.2.2] when we set E = AS, 1 since MS(AS) is contractible. The left-hand isomorphism follows from Zariski descent by 1 considering the standard open cover of PS. 

1 1 We will often abuse notation and write simply PS for the pointed motivic space (MS(PS), ∞) when there is no risk of confusion.

8.3. Motivic P1-spectra.

1 12 1 8.3.1. A motivic spectrum over a scheme S is a motivic PS-spectrum , where we write PS = 1 (MS(PS), ∞) by abuse of notation. 1 We will write SH(S) := SH 1 (S) for the category of motivic P -spectra. PS S

8.3.2. Lemma 8.3.3. The arena SH(S) is stable.

Proof. It is clear that SH(S) is pointed. By [Chap. 1, Corollary 8.2.4], the 1-sphere S1 is invertible in SH(S), which is a necessary and sufficient condition for stability. 

11The cofibre is taken in the unpointed category, and we consider it with its canonical base point. 12This is the “correct” category of motivic spectra, where we will have the full formalism of the six operations. 1 This is why we drop the subscript PS from the notation. 74 1. MOTIVIC SPACES AND SPECTRA

8.3.4. The following fact ensures that [Chap. 1, Lemma 4.1.10] applies, so that we get a canonical symmetric monoidal structure on SH(S).

1 Lemma 8.3.5. The pointed motivic space PS is 3-symmetric.

Proof. It suffices by functoriality to assume that S = Spec(Z). In this case the claim is well-known (see e.g. [Voe98, Lem. 4.4]). 

8.3.6.

Lemma 8.3.7. For each integer n > 0, there are canonical isomorphisms n thomAn/S = Σ 1 , S PS,S n AS /S n thom = Ω 1 . PS,S

Proof. The second isomorphism is the right transpose of the first. For the first, note that by [Chap. 1, Lemma 8.1.7] and [Chap. 1, Corollary 8.2.4], both functors are given by the assignment 1 ⊗n F 7→ F ⊗ (PS) . 

E/S 8.3.8. The adjunction (thomE/S, thom ) gives rise to an adjunction at the level of motivic spectra, which we denote in the same way. We have (see [Chap. 2, Lemma 3.4.12]): Lemma 8.3.9. For every vector bundle E over S, the adjunction E/S (8.9) thomE/S : SH(S)  SH(S) : thom is an equivalence.

8.3.10. Suppose the scheme S is classical, and noetherian of finite Krull dimension (in the classical sense). In this case there is an ordinary triangulated category of motivic spectra constructed by Morel–Voevodsky, which can be viewed as the underlying ordinary category of a stable (∞, 1)-category (see [Rob14] or [Hoy15, Appendix C]). We have: Proposition 8.3.11. If S is a classical noetherian scheme of finite Krull dimension, then the (∞, 1)-category SH(S) coincides with the Morel–Voevodsky (∞, 1)-category of motivic spectra over S.

Proof. This follows directly from [Chap. 1, Proposition 2.4.14], by comparison with the con- struction of [Hoy15]. 

9. The localization theorem

This section is dedicated to the proof of the localization theorem (see [Chap. 1, Para- graph 7.4]). Throughout the section, we let i :Z ,→ S be a closed immersion of schemes, such that the complementary open immersion j :U ,→ S is quasi-compact.

9.1. The space hS(X, t). 9. THE LOCALIZATION THEOREM 75

9.1.1. Given a smooth S-scheme X, let XU := X ×S U denote its base change along j, and XZ := X ×S Z its base change along i. Z We will write hS (X) for the space over S defined by the cocartesian square

hS(XU) hS(X) (9.1)

Z hS(U) hS (X).

Note that there is a canonical isomorphism ∗ Z (9.2) iSpc(hS (X)) = hZ(XZ) ∗ of spaces over Z, since iSpc commutes with colimits. Since colimits in Spc(S) are computed section-wise, we can describe the spaces of sections Z of hS (X) explicitly:

Lemma 9.1.2. Let Y be a smooth S-scheme. If YZ is the empty scheme, then the space Z Γ(Y, hS (X)) is contractible. Otherwise, there is a canonical isomorphism of spaces Z (9.3) Γ(Y, hS (X)) = Γ(Y, hS(X)) = MapsS(Y, X).

9.1.3. Let p :X → S be a smooth morphism. Let t :Z ,→ X be an S-morphism, i.e. a partially defined section of p. Consider the canonical morphism Z Spc ∗ Z Spc (9.4) ε : hS (X) → i∗ iSpc(hS (X)) = i∗ (hZ(XZ)) ∗ Spc induced by the counit of the adjunction (iSpc, i∗ ). Spc The morphism t corresponds by adjunction to a morphism τ : hS(S) → i∗ (hS(XZ)). We define a space hS(X, t) over S as the fibre of ε at the point τ, so that we have a cartesian square

Z hS(X, t) hS (X)

ε

τ Spc hS(S) i∗ (hS(XZ)) of spaces over S.

9.1.4. Over S-schemes that do not vanish on Z, sections of hS(X, t) are S-sections of X extending t. More precisely (recall that limits in Spc(S) are computed section-wise):

Lemma 9.1.5. Let Y be a smooth S-scheme. If YZ is the empty scheme, then the space Γ(Y, hS(X, t)) is contractible. Otherwise, Γ(Y, hS(X, t)) is canonically identified with the fibre of the restriction map

MapsS(Y, X) → MapsZ(YZ, XZ) t at the point defined by the composite YZ → Z ,→ XZ.

In other words, points of the space Γ(Y, hS(X, t)) are pairs (f, α), with f :Y → X an S-morphism and α a commutative triangle

fZ YZ XZ. t

Z 76 1. MOTIVIC SPACES AND SPECTRA

∗ 9.1.6. If p is a smooth morphism, then since pSpc commutes with both limits and colimits, we have: Lemma 9.1.7. Let X be a smooth S-scheme and t :Z ,→ X an S-morphism. If p :T → S is a smooth morphism, then there is a canonical isomorphism of fibred spaces ∗ pSpc(hS(X, t)) = hT(XT, tT), where tT :ZT ,→ XT is obtained from t by base change along p.

9.1.8. Our main result about the fibred space hS(X, t) is as follows: Proposition 9.1.9. Let X be an affine smooth S-scheme. Then for every S-morphism t :Z ,→ X, the space hS(X, t) is motivically contractible.

The proof will occupy the rest of this section.

9.1.10. We first consider the case of vector bundles: Lemma 9.1.11. Let E be a vector bundle over S with zero section s :S ,→ E. Then the space hS(E, sZ) is motivically contractible, where sZ :Z ,→ EZ denotes the base change of s along i :Z ,→ S.

Proof. It suffices to construct an A1-homotopy inverse to the unique morphism

ϕ : hS(E, sZ) → hS(S). The zero section induces a canonical morphism s Z hS(S) −→ hS(E) → hS (E), which induces a canonical morphism

ψ : hS(S) → hS(E, sZ).

It remains to define an A1-homotopy 1 ϑ : hS(AS) × hS(E, sZ) → hS(E, sZ) S between the identity and the composite ψ ◦ ϕ. For each smooth S-scheme Y with YZ =6 ∅, define 1 Γ(Y, ϑ) : Γ(Y, hS(AS)) × Γ(Y, hS(E, sZ)) → Γ(Y, hS(E, sZ)) by the assignment 1 (a :Y → AS, f :Y → E) 7→ (a · f :Y → E). 1 It is clear that this defines the A -homotopy desired. 

9.2. Etale´ base change.

9.2.1. The assignment (X, t) 7→ hS(X, t) is functorial in the following sense. Let (X, t) and (X0, t0) be pairs, with X (resp. X0) a smooth S-scheme, and t :Z ,→ X (resp. t0 :Z ,→ X0) a partially defined section. Suppose f :X0 → X is an S-morphism such that the square 0 t 0 Z XZ

t Z XZ is cartesian. Then there is a canonical morphism of spaces over S 0 0 (9.5) hS(X , t ) → hS(X, t). 9. THE LOCALIZATION THEOREM 77

Lemma 9.2.2. Suppose that (X, t) and (X0, t0) are pairs as above. Let p :X0 → X be an ´etale morphism, such that the above square is cartesian. Then the induced morphism 0 0 ϕ : hS(X , t ) → hS(X, t) is a Nisnevich-local equivalence.

The claim is that the induced morphism of Nisnevich sheaves LNis(ϕ) is invertible. By [Chap. 0, Lemma 2.8.8], it suffices to show that it is 0-truncated (i.e. its diagonal is a monomorphism) and 0-connected (i.e. it is an effective epimorphism and so is its diagonal).

9.2.3. Proof of [Chap. 1, Lemma 9.2.2], step 1. To show that LNis(ϕ) is 0-truncated, it suffices to show that ϕ is 0-truncated (since LNis is exact). For this, it suffices to show that for each smooth S-scheme Y, the induced morphism of spaces of Y-sections Z 0 0 Z Γ(Y, ϕ) : Γ(Y, hS (X , t )) → Γ(Y, hS (X, t)) is 0-truncated.

We may assume YZ is not empty; then this is the morphism induced on fibres in the diagram

Z 0 0 0 0 Γ(Y, hS (X , t )) MapsS(Y, X ) MapsZ(YZ, XZ)

Z Γ(Y, hS (X, t)) MapsS(Y, X) MapsZ(YZ, XZ) Note that the two right-hand vertical morphisms are 0-truncated: p is itself 0-truncated since it is ´etale,and since the Yoneda embedding commutes with limits, the induced morphism 0 hS(X ) → hS(X) is also 0-truncated. It follows that the left-hand vertical morphism is also 0-truncated for each Y, and therefore so is ϕ.

9.2.4. Proof of [Chap. 1, Lemma 9.2.2], step 2. To show that LNis(ϕ) is an effective epimorphism, it suffices to show that for each smooth S-scheme Y (with YZ not empty), any Y-section of Z hS (X, t) can be lifted Nisnevich-locally along ϕ. Z Let f be a Y-section of hS (X, t), i.e. a morphism f :Y → X together with an isomorphism t 0 between fZ and the composite YZ → Z ,→ XZ. Let q :Y → Y denote the base change of p :X0 → X along f: q Y0 Y

g f p X0 X. Then note that −1 0 q (YU) Y q

YU Y is a Nisnevich square. Indeed, the closed immersion YZ ,→ Y is complementary to YU ,→ Y, −1 and it is clear that q (YZ) → YZ is invertible because in the diagram

−1 q (YZ) YZ

Z idZ Z

t t0 0 pZ XZ XZ 78 1. MOTIVIC SPACES AND SPECTRA the lower square and the composite square are cartesian, and hence so is the upper square. Hence it suffices to show that the restriction of f to either component of this Nisnevich Z 0 0 0 Z 0 0 0 0 cover lifts to hS (X , t ). The restriction f|Y lifts to a section of hS (X , t ) given by g :Y → X . Z The restriction f|YU admits a lift trivially: since (YU) ×S Z = ∅, the spaces hS (X, t)(YU) and Z 0 0 hS (X , t )(YU) are both contractible.

9.2.5. Proof of [Chap. 1, Lemma 9.2.2], step 3. It remains to show that the diagonal ∆LNis(ϕ) of LNis(ϕ) is an effective epimorphism, or equivalently that LNis(∆ϕ) is. For each smooth S-scheme Y, the diagonal induces a morphism of spaces Z 0 0 Z 0 0 Z 0 0 Γ(Y, hS (X , t )) → Γ(Y, hS (X , t )) × Γ(Y, hS (X , t )). Z Γ(Y,hS (X,t))

It suffices to show that for each Y (with YZ not empty), any Y-section of the target lifts Nisnevich-locally to a Y-section of the source. Choose a section of the target, given by two Y-sections f :Y → X0 and g :Y → X0, and an identification α : p ◦ f → p ◦ g.

Let Y0 ,→ Y denote the open immersion defined as the equalizer of the pair (f, g). Note that the closed immersion YZ ,→ Y factors through Y0. Hence the open immersions Y0 ,→ Y and YU ,→ Y form a Zariski cover of Y. It is clear that the Y-section (f, g, α) lifts after restriction to Y0 by definition, and after restriction to YU since YU ×S Z = ∅, so the claim follows.

9.3. Reduction to the case of vector bundles. We reduce to the case of vector bundles in two steps: first, we show that partial sections of smooth S-schemes can be lifted Nisnevich-locally to globally defined sections; second, we show using ´etalebase change that smooth S-schemes with globally defined sections can be replaced by their conormal bundles.

9.3.1. The following lemma will allow us to reduce to the situation where the Z-section t lifts to an S-section s :S ,→ X. Lemma 9.3.2. Let p :X → S be a smooth morphism. Given an S-morphism t :Z ,→ X, there exists a Nisnevich square

YU Y (9.6) q j U S such that q factors through p.

Proof. We will construct a commutative square Z Y (9.7)

XZ X. with the following properties: (i) The induced square of underlying classical schemes

Zcl Ycl

(XZ)cl Xcl is cartesian. (ii) The composite morphism Y → X → S is ´etale. 9. THE LOCALIZATION THEOREM 79

Given such a square (9.7), it is clear that we get a Nisnevich square (9.6) as claimed, by taking q to be the composite Y → X → S: indeed, the closed immersion Zcl ,→ S is complementary to j, and the squares

Zcl Ycl Y q

Zcl Scl S are cartesian (the left-hand one by (i), the right-hand one by (ii)). In the classical case, the existence of the square (9.7) is known (this is a non-equivariant version of [Hoy17, Thm. 2.21]). Hence one obtains a cartesian square

Zcl Y0

(XZ)cl Xcl of classical schemes. Then one defines Y by the cocartesian square of closed immersions

Zcl Y0

Z Y.

By [GR16, III.1, Cor. 1.3.5] this is well-defined, and the morphism Y0 ,→ Y is a closed immersion identifying Y0 with the classical scheme underlying Y. The existence of the desired commutative square Z Y

XZ X follows by construction. 

9.3.3. Next, we show that an S-section s :S ,→ X may be approximated by the zero section of its conormal bundle. This is a refinement of [AG15, Prop. 2.1.10]. Lemma 9.3.4. Let p :X → S be a smooth morphism, and assume that p is further affine. If p ∗ admits a section s :S ,→ X, then there exists an S-morphism q :X → Ns to the conormal bundle of s satisfying the following conditions: (i) The commutative diagram S s X q

t ∗ S Ns is cartesian, where t denotes the zero section.

(ii) The morphism s factors through an open immersion j0 :X0 ,→ X with q ◦ j0 ´etale.

∗ Proof. Recall that the conormal bundle Ns is by definition the vector bundle associated to the ∗ ∗ shifted cotangent sheaf Ns = T (S/X)[−1].

Consider the closed immersion scl :Scl ,→ Xcl of underlying classical schemes. Let I denote ∗ its defining quasi-coherent sheaf of ideals, and Nscl = i (Ns) its conormal bundle, where i is the closed immersion Scl ,→ S. 80 1. MOTIVIC SPACES AND SPECTRA

The epimorphism (p ) (I) → N∗ admits a section, since N∗ is projective, so that one cl ∗ scl scl obtains a morphism N∗ → (p ) (O ). This lifts to a morphism N∗ → p (O ), corresponding scl cl ∗ Xcl s ∗ X to a morphism of OS-algebras

∗ ϕ : SymOS (Ns ) → p∗(OX).

Then it is clear that the commutative square of OS-algebras

∗ ϑ Ns OS ϕ

σ p∗(OX) OS is cocartesian. ∗ We let q :X → Ns be the morphism of S-schemes corresponding to ϕ.

For (ii), let j0 be the ´etale locus of q. To show that s factors through j0, it is sufficient to ∗ ∗ ∗ note that s (T (X/Ns)) = 0. 

9.4. Motivic contractibility of hS(X, t).

9.4.1. Let X be an affine smooth S-scheme, with structural morphism p :X → S. Recall the statement of [Chap. 1, Proposition 9.1.9]: we want to show that for any S-morphism t :Z ,→ X, the fibred space hS(X, t) is motivically contractible.

9.4.2. By [Chap. 1, Lemma 9.3.2] there exists a Nisnevich square

YU Y (9.8) q j U S where q factors through p :X → S. It suffices then by the Nisnevich separation property ([Chap. ∗ ∗ 0 1, Proposition 6.1.6]) to show that j hS(X, t) and q hS(X, t) = hY(Y ×S X, t ) are contractible, 0 where t :YZ ,→ (Y ×S X)Z is the base change of t.

∗ 9.4.3. The case of j hS(X, t) is clear, since j is complementary to i :Z ,→ S.

∗ 00 9.4.4. For q hS(X, t), note that by construction there exists a section t :Y ,→ Y ×S X which lifts t0 (since q factors through X):

(Y ×S X)Z Y ×S X

t0 t00

YZ Y

Hence by [Chap. 1, Lemma 9.3.4], [Chap. 1, Lemma 9.2.2] and [Chap. 1, Lemma 9.1.11], we have motivic equivalences

0 ∗ hS(Y × X, t ) = hS(Nt00 , z) = hS(S), S ∗ where Nt00 is the conormal bundle, and z is the base change of its zero section. 9. THE LOCALIZATION THEOREM 81

9.5. Proof of the localization theorem. We conclude this section by proving the localization theorem, using [Chap. 1, Proposition 9.1.9]. Recall that our goal is to show that the canonical morphism ∗ (9.9) F t MS(U) → i∗i (F) ∗ j]j (F) is invertible for each motivic space F over S.

9.5.1. First, note that we may reduce to the case where F is a motivic localization MS(X) of an affine smooth S-scheme X. Indeed, we have seen that the category H(S) is generated under sifted colimits by such objects ([Chap. 1, Proposition 2.4.12]) and that each of the functors j], ∗ ∗ j , i , and i∗ commutes with contractible colimits ([Chap. 1, Proposition 7.3.2]). In this case the morphism (9.9) is canonically identified with the morphism

(9.10) MS(X) t MS(U) → i∗ MS(XZ) MS(XU) where we write XU = X ×S U and XZ = X ×S Z.

Z 9.5.2. Note that the source of the morphism (9.10) is the motivic localization of the space hS (X), H Spc and that the target i∗ (MZ(XZ)) is the motivic localization of i∗ (hZ(XZ)). Hence it suffices to show that the morphism Z Spc (9.11) hS (X) → i∗ hZ(XZ) is a motivic equivalence.

9.5.3. By universality of colimits ([Chap. 1, Proposition 2.4.10]), it suffices to show that for every Spc smooth S-scheme Y and every morphism hS(Y) → i∗ hZ(XZ), corresponding to an S-morphism t :Z → X, the base change Z (9.12) hS (X) × hS(Y) → hS(Y) Spc i∗ hZ(XZ) is invertible.

Spc 9.5.4. Let p :Y → S be the structural morphism of Y. Then since hS(Y) = p] hY(Y), one sees that (9.12) is identified, by the smooth projection formula ([Chap. 1, Lemma 6.3.3]), with a morphism Spc ∗ Z Spc (9.13) p] (pSpc hS (X) × hY(Y)) → p] hY(Y). ∗ Spc pSpc i∗ hZ(XZ)

∗ ∗ 9.5.5. Note that we have p i∗ = k∗q ([Chap. 1, Proposition 6.2.2]), where k (resp. q) is the base change of i (resp. p) along p (resp. i). Hence the morphism (9.13) is identified with the image by p] of

YZ (9.14) hY (X × Y) × hY(Y) → hY(Y). S Spc k∗ hYZ ((X ×S Y)Z)

9.5.6. The source of the morphism (9.14) is nothing else than the space hY(X ×S Y, tY), where tY :Z ×S Y → X ×S Y is the base change of t along p. Hence we conclude by [Chap. 1, Proposition 9.1.9].

CHAPTER 2

The formalism of six operations

1. Introduction 84 1.1. The six operations 84 1.2. The 2-category of correspondences 85 1.3. Biadjointable categories of coefficients 88 1.4. Motivic categories of coefficients 89 1.5. Organization of this chapter 89 2. Categories of coefficients 91 2.1. Categories of coefficients 91 2.2. Left-adjointability 92 2.3. Right-adjointability 93 2.4. Biadjointability 94 3. Motivic categories of coefficients 96 3.1. Premotivic categories of coefficients 96 3.2. Homotopy invariance 96 3.3. Localization 97 3.4. Thom stability 101 3.5. Motivic categories of coefficients 104 3.6. Comparison with the axiomatic of Cisinski–Deglise 105 3.7. Biadjointability of motivic categories of coefficients 107 4. The formalism of six operations 111 4.1. Schematic correspondences 111 4.2. The extension theorem 112 4.3. Biadjointability on schematic arrows 113 4.4. The extension theorem, proof 116 5. Example: the stable motivic homotopy category 118 5.1. The functor SH∗ 118 ∗ 5.2. The 2-functor SH! 119

83 84 2. THE FORMALISM OF SIX OPERATIONS

1. Introduction

One of the main goals of this document is to construct the formalism of six operations for the stable motivic homotopy category, in the setting of derived algebraic geometry. In this chapter, we develop an axiomatic approach to the construction of this formalism.

1.1. The six operations.

1.1.1. The principle of Grothendieck’s yoga of six operations is that cohomology theories in algebraic geometry (e.g. `-adic cohomology or de Rham cohomology), generally come with some category of coefficients (e.g. `-adic sheaves or D-modules). These categories of coefficients ∗ ! come with the six operations (f , f∗, f!, f , ⊗, Hom), which categorify the standard properties of cohomology theories like K¨unnethformulas, Poincar´eduality and Gysin maps.

1.1.2. A category of coefficients satisfying the formalism of six operations consists roughly of the following data: (1) For each (derived) scheme S, a closed symmetric monoidal (∞, 1)-category D(S). (2-a) For each morphism f :S0 → S, a functor of inverse image f ∗ : D(S) → D(S0) which is symmetric monoidal. (2-b) For each morphism f :S0 → S, a functor of direct image 0 f∗ : D(S ) → D(S), right adjoint to f ∗. (3-a) For any morphism f :S0 → S which is separated of finite type, a functor of exceptional direct image (or direct image with compact support) 0 f! : D(S ) → D(S).

(3-b) For any morphism f :S0 → S which is separated of finite type, a functor of exceptional inverse image f ! : D(S) → D(S0), right adjoint to f!. (4) For any separated morphism of finite type f, a functorial 2-morphism

αf : f! → f∗ which is invertible when f is proper. (5) For any cartesian square of schemes

f 0 T0 S0

g0 g f T S with g and g0 separated of finite type, isomorphisms (base change formulas)

∗ ∼ 0 0 ∗ f g! −→ (g )!(f ) ,

0 0 ! ∼ ! (f )∗(g ) −→ g f∗. 1. INTRODUCTION 85

(6) For any morphism f :S0 → S which is separated of finite type, isomorphisms (projection formulas)

∼ ∗ 0 F ⊗S f!(G) −→ f!(f (F) ⊗S0 G), (F ∈ D(S), G ∈ D(S )) ∗ ! ∼ ! HomS0 (f (F), f (G)) −→ f HomS(F, G), (F, G ∈ D(S)) ! ∼ 0 f∗HomS0 (F, f (G)) −→ HomS(f!(F), G). (F ∈ D(S ), G ∈ D(S))

1.1.3. All this data should come with a homotopy-coherent system of compatibilities. ∗ ∗ ∗ For example, there should be connection isomorphisms (g ◦ f) → f g and (g ◦ f)! → g!f! for any two composable morphisms of schemes f and g, with compatibilities between such isomorphisms. The base change and projection formulas should also come with compatibilities with these connection isomorphisms.

1.2. The 2-category of correspondences.

1.2.1. As a starting point, we can coherently encode the individual operations and the adjunctions ∗ ! (f , f∗), (f!, f ), (⊗, Hom) by the data of a symmetric monoidal functor D∗ :(Sch)op → Arena, and a functor

D! : Sch → Arena which take the same values on objects. Here Arena denotes the (∞, 1)-category of arenas (see [Chap. 0, Paragraph 2.6]). ∗ We write D(S) := D (S) = D!(S) for the (∞, 1)-category associated to each scheme S. Since D∗ is symmetric monoidal and preserves commutative monoid objects, there is an induced structure of commutative monoid in Arena on D(S) (since every scheme S has a canonical structure of cocommutative comonoid via the diagonal morphism). Recall that commutative monoids in Arena are by definition symmetric monoidal arenas, i.e. arenas with a symmetric monoidal structure such that the bifunctor − ⊗ − admits a right adjoint Hom(−, −). Hence we have encoded the operations (⊗, Hom). For a morphism of schemes f, we write f ∗ := D∗(f) for the induced morphism of arenas, and f∗ for its right adjoint. We have a functor

D∗ : Sch → (∞, 1)-Cat obtained from D∗ by passing to right adjoints (see [Chap. 0, Paragraph 3.2]). Hence we have ∗ coherently encoded the operations (f , f∗). ! Similarly, the operations (f!, f ) are encoded by D! and the functor D! :(Sch)op → (∞, 1)-Cat obtained by passage to right adjoints.

1.2.2. In order to coherently encode base change formulas, we will follow the approach of D. Gaitsgory and N. Rozenblyum [GR16], using the (∞, 2)-category of correspondences. proper In [GR16, Book V.1] the authors define a 2-category Corr(Sch)sep;all , which can be described informally as follows. Its objects are schemes. The 1-morphisms S0 → S are correspondences, i.e. 86 2. THE FORMALISM OF SIX OPERATIONS diagrams

f T S0 (1.1) g S where g is separated of finite type. The 2-morphisms from a correspondence (T0, f 0, g0) to a correspondence (T, f, g) are diagrams

0 T0 f h

f g0 T S0 g S where h :T0 → T is proper. Composition of 1-morphisms is given by forming fibred products, and composition of 2-morphisms is defined in the evident way.

∗ 1.2.3. The two functors D and D!, together with the base change formulas and all the relevant coherences, can be encoded by the datum of a symmetric monoidal functor

∗ proper 2-op D! : Corr(Sch)sep;all → (Arena) on the 2-category of correspondences. ∗ Given a scheme S, D! sends it to a symmetric monoidal arena D(S). In particular, the closed symmetric monoidal structure encodes the operations (⊗, Hom). ∗ Given a correspondence of the form (1.1), D! sends it to the composite ∗ 0 g!f : D(S ) → D(S).

∗ Hence by restricting D! to the full sub-1-category of correspondences of the form (1.1) where the vertical morphisms are identity, we recover the functor D∗ encoding the operations ∗ (f , f∗). ∗ By restricting D! to the full sub-1-category of correspondences where the horizontal mor- ! phisms are identity, we recover the functor D! encoding the operations (f!, f ).

1.2.4. Suppose there is a cartesian square of schemes

f 0 T0 S0

(1.2) g0 g f T S with g and g0 separated of finite type. The composite of the two correspondences

f S0 id S0 T S

g and id S T 1. INTRODUCTION 87 is the correspondence f 0 T0 S0 id S0

g0 g f T S

id T ∗ Hence applying the functor D! , we obtain the base change isomorphism ∗ ∼ 0 0 ∗ f g! −→ (g )!(f ) .

1.2.5. Let f :T → S be a separated morphism of finite type. Then its diagonal ∆T/S :T → T ×S T is a closed immersion (hence a fortiori proper), and therefore defines a morphism of correspondences (T, idT, idT) → (T ×S T, pr1, pr2) = (T, f, idT) ◦ (T, idT, f) given by the commutative diagram T ∆T/S

pr1 T ×S T T T

pr2 f f T S

T ∗ Applying the 2-functor D! (which is contravariant on 2-morphisms), we obtain a canonical ∗ functorial 2-morphism η : f f! → idT, or by adjunction, a canonical functorial 2-morphism

αf : f! → f∗.

If f is further proper, then one has a morphism of correspondences (T, f, f) → (S, idS, idS) given by the commutative diagram T f f

f S S

S ∗ This gives rise to a canonical functorial 2-morphism ε : id → f!f . ∗ One checks that the 2-morphisms (η, ε) define an adjunction (f , f!). In particular, we see that the 2-morphism αf : f! → f∗ is invertible when f is proper.

proper 2-op 1.2.6. Finally, the functor Corr(Sch)sep;all → (Arena) needs to be modified in order to encode the projection formulas. For this we make use of an observation implicit in [LZ12]: the projection formula ∗ ∼ f!(F ⊗T f (G)) −→ f!(F) ⊗S G is expressing nothing else than the D(S)-linearity of the functor f! : D(T) → D(S). In other words, it expresses the fact that f! is a morphism of D(S)-module categories, where D(T) is given the structure of D(S)-module via the symmetric monoidal functor f ∗. proper 2-op With this in mind, we modify our functor Corr(Sch)sep;all → (Arena) as follows. 88 2. THE FORMALISM OF SIX OPERATIONS

Firstly, we want to change the target from Arena to the 2-category Arenamod of pairs (O, C) with O a symmetric monoidal arena and C an O-module arena. Morphisms (O, C) → (O0, C0) are given by pairs (u, v) with u : O → O0 a symmetric monoidal morphism of arenas and v : C → C0 a morphism of O-module arenas, where C0 is viewed as a O-module via the functor u. Given a morphism f :T → S, we want to view f ∗ as a morphism (1.3) (D(S), D(S)) → (D(S), D(T)) and f! as a morphism (1.4) (D(S), D(T)) → (D(S), D(S)).

For this, let Arrsch denote the category of schematic arrows, i.e. morphisms of schemes. Then we consider the functor ∗ op DArrsch :(Arrsch) → Arenamod, where Arenamod denotes the underlying 1-category of Arenamod, defined on objects and morphisms as follows: It sends an arrow f :T → S to the pair (D(S), D(T)), where D(T) is viewed as an D(S)-module via the monoidal functor f ∗ : D(S) → D(T). It sends a morphism of arrows α : f 0 → f, given by a commutative square

f 0 T0 S0

g0 g f T S, to a morphism of pairs (D(S), D(T)) → (D(S0), D(T0)) given by the monoidal functor g∗ : D(S) → D(S0) and the D(S)-linear functor (g0)∗ : D(T) → D(T0).

∗ This functor DArrsch encodes the desired morphism (1.3) as the image of the morphism of schematic arrows f → idS given by the commutative square f T S f S S.

Arrsch Similarly we want to encode the morphism (1.4) by a functor D! : Arrsch → Arenamod.

∗ 1.2.7. Hence we need to change the source of our functor D! to the 2-category proper Corr(Arrsch)sep;all , proper of correspondences of schematic arrows, which can be defined in a similar way as Corr(Sch)sep;all . In summary, the full formalism of six operations will be encoded by a symmetric monoidal 2-functor proper 2-op Corr(Arrsch)sep;all → (Arenamod) .

1.3. Biadjointable categories of coefficients. 1. INTRODUCTION 89

1.3.1. We will refer to the datum of a symmetric monoidal functor D∗ :(Sch)op → Arena simply as a category of coefficients. We say that D∗ is (open, proper)-biadjointable ([Chap. 2, Definition 2.4.4]) if the following conditions hold: (1) D∗ is left-adjointable along open immersions, i.e. for all open immersions j, the functor ∗ j admits a left adjoint j] which satisfies base change and projection formulas. (2) D∗ is right-adjointable along proper morphisms, i.e. for all proper morphisms f, the ! functor f∗ admits a right adjoint f , and f∗ satisfies base change and projection formulas. ∗ (3) D satisfies a base change formula expressing commutativity of the operations j] and f∗ (j an open immersion, f proper).

1.3.2. Given an (open, proper)-biadjointable category of coefficients, [Chap. 2, Theorem 4.2.2] says that D∗ can be extended to a 2-functor ∗ proper 2-op D! : Corr(Arrsch)sep;all → (Arena) encoding a formalism of six operations. This functor is constructed by applying the technology of [GR16]. The basic idea, due to P. ! Deligne [SGA 4, Exp. XVII], is that the exceptional operations (f!, f ) should be defined by choosing a compactification of f, i.e. a factorization of f as an open immersion j followed by a proper morphism g, and setting: ! ∗ ! f! = g∗j!, f = j g .

More precisely, f! should be defined as the colimit of the contravariant functor (j, p) 7→ p∗j! on the category of compactifications of f.

1.4. Motivic categories of coefficients.

1.4.1. In order to obtain the six functor formalism for the stable motivic homotopy category, we need to demonstrate (open, proper)-biadjointability. What we have by construction is a category of coefficients which is left-adjointable with ∗ respect to smooth morphisms. That is, we have functors p] left adjoint to p for all smooth morphisms p, satisfying base change and projection formulas. [Chap. 2, Theorem 3.5.4] identifies a set of sufficient conditions for a category D∗, left- adjointable along smooth, to be (open, proper)-biadjointable. We say that D∗ is motivic ([Chap. 2, Definition 3.5.2]) if it satisfies these conditions. In particular, we can then apply [Chap. 2, Theorem 4.2.2] to obtain a full formalism of six operations on D∗. For the stable motivic homotopy category, these conditions have been verified in Chapter 1.

1.5. Organization of this chapter. In Sect. 2 we introduce the notion of biadjointable category of coefficients. In Sect. 3 we introduce the property motivic for a category of coefficients, and show that any motivic category of coefficients is biadjointable. For classical schemes, the analogue of this result (in the language of triangulated categories) is due to J. Ayoub [Ayo07] (cf. [CD09]). We assume his result in our proof. In Sect. 4 we show that any biadjointable category of coefficients extends to a functor on the category of correspondences, encoding the formalism of six operations. This result is a direct 90 2. THE FORMALISM OF SIX OPERATIONS application of [GR16]. Our only departure from loc. cit. is the introduction of schematic arrows in order to encode projection formulas. In Sect. 5 we return to our main example of interest, the category of motivic spectra. We construct this as a motivic category of coefficients and deduce the existence of the formalism of six operations. 2. CATEGORIES OF COEFFICIENTS 91

2. Categories of coefficients

2.1. Categories of coefficients.

2.1.1. Recall that Arena denotes the symmetric monoidal (∞, 1)-category of arenas and colimit- preserving functors (see [Chap. 0, Paragraph 2.6]), and Arenamon denotes the (∞, 1)-category of symmetric monoidal arenas, which are by definition commutative monoids in Arena. For the duration of this section, we fix an (∞, 1)-category C which admits fibred products. Note that the cartesian monoidal structure on C induces a canonical symmetric monoidal structure on (C)op.

Definition 2.1.2. A category of coefficients (defined on C) is a symmetric monoidal functor

D∗ :(C)op → Arena.

Given a category of coefficients D∗, we will write

D(S) := D∗(S) for the arena associated to an object S ∈ C. We will write ∅S (resp. eS) for the initial (resp. terminal) object of D(S). We will often refer to the objects of D(S) as sheaves on S. For a morphism f :T → S, we will write

f ∗ := D∗(f): D(S) → D(T) for the induced functor, which we call the functor of inverse image along f. It is cocontinuous, and admits (by the adjoint functor theorem) a right adjoint

f∗ : D(T) → D(S) which we call the functor of direct image along f.

2.1.3. By passing to right adjoints (see [Chap. 0, Paragraph 3.2]), D∗ gives rise to a unique functor

D∗ : C → (∞, 1)-Cat such that each functor

D∗(f): D(T) → D(S) is the right adjoint f∗.

2.1.4. Since the functor D∗ underlying a category of coefficients is symmetric monoidal, it sends cocommutative comonoids in C to commutative monoids in Arena. Note that every object S ∈ C has a canonical structure of cocommutative comonoid (with respect to the cartesian monoidal structure). Hence for each object S ∈ C, the arena D(S) has a canonical symmetric monoidal structure, and for each morphism f in C, the inverse image functor f ∗ has a canonical symmetric monoidal structure (giving by adjunction a lax monoidal structure on its right adjoint f∗).

We will write ⊗S for the monoidal product of D(S), 1S for the monoidal unit, and HomS for the internal hom. 92 2. THE FORMALISM OF SIX OPERATIONS

2.1.5. Dually, suppose we are given a functor (not necessarily symmetric monoidal)

D! : C → Arena.

We will write D(S) := D!(S) for the arena associated to an object S ∈ C. For each morphism f :T → S in C, we write f! := D!(f): D(T) → D(S) for the induced functor, which commutes with colimits and admits a right adjoint f !. As above, we can pass to right adjoints to obtain a functor D! :(C)op → Arena.

2.1.6. For future use, we make the following definition: Definition 2.1.7. A category of coefficients D∗ is pointed (resp. S1-stable, compactly gener- ated) if the functor D∗ :(C)op → Arena factors through the full subcategory spanned by pointed (resp. stable, compactly generated) arenas.

2.2. Left-adjointability.

2.2.1. Let us fix a class left of left-admissible morphisms in C, containing all isomorphisms, closed under composition and base change, and satisfying the 2-out-of-3 property. Let Cleft denote the (non-full) subcategory of C spanned by left-admissible morphisms. When C is the category of schemes, we will typically have left = open, the class of (quasi-compact) open immersions, or left = smooth, the class of smooth morphisms (of finite presentation). Definition 2.2.2. We say that the category of coefficients D∗ is weakly left-adjointable along a morphism p :T → S if it satisfies the following property: p ∗ (Adj ) The functor p admits a left adjoint p]. We say that D∗ is weakly left-adjointable along the class left if it satisfies the following property: (Adjleft ) For every left-admissible morphism p, the property (Adjp) holds.

Note that if D∗ is weakly left-adjointable along left, then one obtains a canonical functor left D] : C → Arena by passage to left adjoints (see [Chap. 0, Paragraph 3.2].

2.2.3. Recall the notion of adjointability of squares from [Chap. 0, Paragraph 3.3]. Definition 2.2.4. The category of coefficients D∗ satisfies left base change along a morphism p :S0 → S if it is weakly left-adjointable along left, and the following property holds: (BCp) For all cartesian squares Θ

f 0 T0 S0

(2.1) p0 p f T S, the induced commutative square Θ∗

f ∗ D(S) D(T)

(2.2) p∗ (p0)∗ (f 0)∗ D(S0) D(T0). 2. CATEGORIES OF COEFFICIENTS 93 is vertically left-adjointable. We say that D∗ satisfies left base change along the class left if it is weakly left-adjointable along left, and the following property holds: (BCleft ) For every left-admissible morphism p, the property (BCp) holds.

In other words, D∗ satisfies left base change along a morphism p if for every such cartesian square Θ, the exchange 2-morphism 0 0 ∗ ∗ (p )](f ) → f p] is invertible.

2.2.5. For any morphism f :T → S, the symmetric monoidal functor f ∗ : D(S) → D(T) ∗ gives D(T) a structure of D(S)-module category. If f] is left adjoint to f , then by [Chap. 0, Lemma 2.7.7] it admits a canonical structure of colax morphism of D(S)-modules. In particular there are canonical morphisms ∗ f](F ⊗T f (G)) → f](F) ⊗S G (F ∈ D(T), G ∈ D(S)). Definition 2.2.6. The category of coefficients D∗ satisfies the left projection formula along a morphism p :T → S if it is weakly left-adjointable along p, and the following property holds: p (Proj ) The colax morphism of D(S)-modules p] is strict. We say that D∗ satisfies the left projection formula along the class left if it is weakly left-adjointable along left, and the following property holds: (Projleft ) For every left-admissible morphism p, the property (Projleft ) holds.

In other words, D∗ satisfies the left projection formula along p :T → S if the canonical morphisms ∗ (2.3) p](F ⊗T p (G)) → p](F) ⊗S G (F ∈ D(T), G ∈ D(S)) are invertible.

2.2.7. Definition 2.2.8. A category of coefficients D∗ is left-adjointable along the class left if it is weakly left-adjointable along left (Adjleft ), satisfies left base change along left (BCleft ), and satisfies the left projection formula along left (Projleft ).

2.3. Right-adjointability.

2.3.1. Let us fix a class right of right-admissible morphisms in C, containing all isomorphisms, and closed under composition and base change. Let Cright denote the (non-full) subcategory of C spanned by right-admissible morphisms. When C is the category of schemes, we will typically have right = closed, the class of closed immersions, or right = proper, the class of proper morphisms.

2.3.2. The following definitions are dual to the definitions in [Chap. 2, Paragraph 2.2]. Definition 2.3.3. A category of coefficients D∗ is weakly right-adjointable along a morphism q if it satisfies the following property:

(Adjq) The direct image functor q∗ admits a right adjoint. We say that D∗ is weakly right-adjointable along a class right if it satisfies the following property: 94 2. THE FORMALISM OF SIX OPERATIONS

(Adjright ) For each right-admissible morphism q, the property (Adjq) holds. Definition 2.3.4. A category of coefficients D∗ satisfies right base change along a morphism q :S0 → S if the following property holds:

(BCright ) For all cartesian squares Θ in C

f 0 T0 S0

q0 q f T S with q and q0 right-admissible, the induced commutative square Θ∗

f ∗ D(S) D(T)

q∗ (q0)∗ (f 0)∗ D(S0) D(T0). is vertically right-adjointable. We say that D∗ satisfies right base change along the class right if the following property holds:

(BCright ) For every right-admissible morphism q, the property (BCq) holds.

In other words, D∗ satisfies right base change along a morphism q if for every such cartesian square Θ, the exchange 2-morphism ∗ 0 0 ∗ f p∗ → (p )∗(f ) is invertible. Definition 2.3.5. A category of coefficients D∗ satisfies the right projection formula along a morphism q :T → S if the following property holds:

(Projq) The canonical structure of lax morphism of D(S)-module arenas on q∗ is strict. We say that D∗ satisfies the right projection formula along the class right if the following property holds:

(Projright ) For every right-admissible morphism q, the property (Projq) holds.

In other words, D∗ satisfies the right projection formula along q :T → S if the canonical morphisms ∗ (2.4) q∗(F ⊗T q (G)) → q∗(F) ⊗S G (F ∈ D(T), G ∈ D(S)). are invertible. Definition 2.3.6. A category of coefficients D∗ is right-adjointable along the class right if it is weakly right-adjointable along right (Adjright ), satisfies right base change along right (BCright ), and satisfies the right projection formula along right (Projright ).

2.4. Biadjointability.

2.4.1. Let D∗ be a category of coefficients which is left-adjointable along left. For every cartesian square Θ in C

q0 T0 S0

(2.5) p0 p q T S 2. CATEGORIES OF COEFFICIENTS 95 with p and p0 left-admissible, we have by left base change along left (BCleft ) a commutative square

q∗ D(S) D(T)

0 (2.6) p] (p )] (q0)∗ D(S0) D(T0).

Suppose that D∗ is also right-adjointable along right and that q and q0 are right-admissible. Then one can ask whether the above square is horizontally right-adjointable, i.e. whether the square D(S) D(T) q∗

0 (2.7) p] (p )] 0 0 D(S ) 0 D(T ) (q )∗ commutes via the 2-morphism 0 ∗ 0 0 0 ∗ 0 0 (2.8) p](q )∗ → q∗q p](q )∗ ' q∗(p )](q ) (q )∗ → q∗(p )]. Definition 2.4.2. The category of coefficients D∗ satisfies bidirectional base change along the pair (left, right) if it satisfies left base change along left, right base change along right, and the following property holds: left (BCright ) For all cartesian squares Θ in C

q0 T0 S0

p0 p q T S with p and p0 left-admissible (resp. q and q0 right-admissible), the square (2.7) commutes.

In other words, we require that for all cartesian squares Θ as above, the 2-morphism (2.8) is invertible.

2.4.3. Finally, we define: Definition 2.4.4. A category of coefficients D∗ is (left, right)-biadjointable if it left-adjointable along left, right-adjointable along right, and satisfies bidirectional base change along (left, right) left (BCright ).

In [Chap. 2, Sect. 4], we will show that on the category of schemes, any (open, proper)- biadjointable coefficient system can be extended to a full formalism of six operations, where open and proper denote the classes of open immersions and proper morphisms, respectively. 96 2. THE FORMALISM OF SIX OPERATIONS

3. Motivic categories of coefficients

3.1. Premotivic categories of coefficients.

3.1.1. For the duration of this section, Sch will denote a full subcategory of the category of schemes, which is stable under coproducts and fibred products; the term scheme will refer to objects of Sch. All categories of coefficients we consider will be defined on Sch. Let smooth (resp. open) denote the class of smooth morphisms of finite presentation (resp. of quasi-compact open immersions). Definition 3.1.2. A category of coefficients D∗ is premotivic if it is left-adjointable along smooth.

3.1.3. The following mild condition on D∗ will always be satisfied in practice: Definition 3.1.4. A category of coefficients D∗ is additive if, for any finite family of schemes (Sα)α, the canonical functor Y D(tαSα) → D(Sα) α is an equivalence.

In particular, we require that the category D(∅) is trivial.

3.1.5. Let τ be a Grothendieck topology on the category Sch. Definition 3.1.6. A category of coefficients D∗ is R-separated, for a τ-covering sieve R, if it satisfies the following property: ∗ (SepR) The family of functors f , with f a morphism in R, is conservative. We say that D∗ is τ-separated if it satisfies the following property: ∗ (Sepτ ) For every τ-covering sieve R, D satisfies the property (SepR).

Remark 3.1.7. Note that the sieve R is the τ-sieve generated by a family of morphisms (fi)i, ∗ ∗ then D is R-separated if and only if the family of functors (fi )i is conservative. ∗ In particular, if the topology τ is generated by a pre-topology τ0, then D is τ-separated if ∗ and only if for every τ0-covering family (fi)i, the family of functors (fi )i is conservative.

3.2. Homotopy invariance.

3.2.1. Let D∗ be a premotivic category of coefficients. Definition 3.2.2. A category of coefficients D∗ is called homotopy invariant if it satisfies the following property: (Htp) For every scheme S, and every vector bundle p :E → S, the inverse image functor p∗ : D(S) → D(E) is fully faithful.

The following observation is a basic property of adjunctions: Lemma 3.2.3. If D∗ is premotivic, then the following conditions are equivalent: (i) D∗ is homotopy invariant. ∗ (ii) For every scheme S, and every vector bundle p :E → S, the counit morphism p]p → id is invertible. 3. MOTIVIC CATEGORIES OF COEFFICIENTS 97

The next lemma says that, in the presence of Zariski separation, it suffices to consider the 1 projections p : AS → S. Lemma 3.2.4. Let D∗ be a premotivic category of coefficients. Consider the following conditions: (i) D∗ is homotopy invariant. ∗ (ii) For every scheme S and integer n > 0, the inverse image p , along the projection n p : AS → S, is fully faithful. ∗ 1 (iii) For every scheme S, the inverse image p , along the projection p : AS → S, is fully faithful. Then the conditions (ii) and (iii) are equivalent. If D∗ satisfies the property of Zariski separation, then (i) and (ii) are also equivalent.

n Proof. For n > 0, the projection p : AS → S can be written as a composite of n projections of 1 the form AX → X, for a scheme X. This demonstrates the equivalence between the conditions (ii) and (iii). For the equivalence between (i) and (ii), assume that D∗ satisfies Zariski separation. This property (together with the smooth base change formula) implies that the condition of the ∗ counit p]p → id to be invertible, is Zariski-local in S. The conclusion follows. 

3.3. Localization. In this paragraph we introduce the localization property, and restate some results of [CD09, §2.3] in our setting.

3.3.1. Let i :Z ,→ S be a closed immersion with quasi-compact open complement j :U ,→ S. We deduce some immediate consequences of the base change property (BCsmooth) in this situation. Considering the commutative square U U j j U S, which is cartesian because j is a monomorphism, we get: Lemma 3.3.2. Let D∗ be a premotivic category of coefficients. Then for any quasi-compact open immersion j :U ,→ S, the canonical morphisms ∗ id → j j], ∗ j j∗ → id are invertible.

In other words, the functors j] and j∗ are fully faithful.

3.3.3. Consider the exchange 2-morphism (2.8)

(3.1) γj : j] → j∗. associated to the cartesian square U U

U S.

We have: 98 2. THE FORMALISM OF SIX OPERATIONS

Lemma 3.3.4. Let D∗ be an S1-stable category of coefficients. Let S be a scheme with connected Q components (Sα)α. Suppose that D(∅) = 0 and the canonical functor D(S) → α D(Sα) is conservative. Then the following conditions are true:

(i) The canonical 2-morphism γjα is invertible for each α, where jα denotes the inclusion Sα ,→ S. Q (ii) The canonical functor D(S) → α D(Sα) is an equivalence.

Proof. For the first claim, it suffices by assumption to show that γjα is invertible after applying ∗ any of the functors (jβ) . For α = β, this follows from the fact that (jα)] and (jα)∗ are fully faithful by [Chap. 2, Lemma 3.3.2]. For α 6= β this follows by left base change, using the assumption that D(∅) = 0. For the second claim, we note that the functor in question admits a left adjoint Y D(Sα) → D(S) α given by the assignment (Fα)α 7→ ⊕α(jα)](Fα). One checks that the unit 2-morphism is invertible, because (jα)] are fully faithful ([Chap. 2, Lemma 3.3.2]), and the counit 2-morphism ∗ is invertible, which can be checked after application of each the functors (jβ) . 

In particular, we obtain: Corollary 3.3.5. If D∗ is S1-stable and Zariski separated, then it is additive if and only if D(∅) = 0.

3.3.6. Considering the cartesian square

∅ Z i j U S, we get:

Lemma 3.3.7. Let D∗ be a premotivic category of coefficients. Suppose that the category D(∅) is trivial1. Then for any closed immersion i :Z ,→ S with quasi-compact open complement j :U ,→ S, the canonical morphisms

∗ ∅Z → i j](F)(F ∈ D(U)), ∗ j i∗(F) → eU (F ∈ D(Z)) are invertible.

3.3.8. Consider the canonical commutative square

∗ j]j (F) F

∗ ∗ ∗ j]j i∗i (F) i∗i (F) for any object F ∈ D(S).

1By this we mean the terminal category, with a unique object and a unique morphism. 3. MOTIVIC CATEGORIES OF COEFFICIENTS 99

By [Chap. 2, Lemma 3.3.7] this induces a canonical commutative square

∗ j]j (F) F (3.2)

∗ j](eU) i∗i (F) which we call the localization square associated to the pair (i, j). Definition 3.3.9. Let i be a closed immersion with quasi-compact open complement. The ∗ premotivic category of coefficients D satisfies the localization property (Loci) with respect to i, if the following conditions hold: (i) The category D(∅) is trivial.

(ii) The functor i∗ is fully faithful. (iii) For every object F ∈ D(S), the localization square (3.2) is cocartesian. ∗ We say that D satisfies the localization property (Loc) if it satisfies (Loci) for all closed immersions i with quasi-compact open complement.

3.3.10. Let D∗ be a premotivic category of coefficients satisfying S1-stability. In this case, condition (iii) of the localization property is equivalent to exactness of the triangle ∗ ∗ (3.3) j]j (F) → F → i∗i (F)(F ∈ D(S)).

We also have the following reformulation: Lemma 3.3.11. Let D∗ be a premotivic category of coefficients satisfying S1-stability. Then the following conditions are equivalent: (i) D∗ satisfies the localization property. (ii) The category D(∅) is trivial, and for any closed immersion i :Z ,→ S with quasi-compact ∗ ∗ open complement j :U ,→ S, the functor i∗ is fully faithful and the pair (j , i ) is conservative.

Proof. Suppose D∗ satisfies the localization property. To show that (j∗, i∗) is conservative, it suffices by S1-stability to show that if F is a object of D(S) such that j∗(F) = 0 and i∗(F) = 0, then F = 0. This follows immediately from the exactness of the triangle (3.3). Conversely, suppose condition (ii) holds. It suffices to show that, for each object F ∈ D(S), the canonical morphism ϕ in the commutative triangle

η ∗ F i∗i (F) ψ ϕ

∗ Cofib(j]j (F) → F) is invertible. By assumption, it suffices to show it it becomes invertible after applying either of the functors j∗ or i∗. Applying j∗, we get ∗ ∗ j Cofib(j]j (F) → F) = 0 by [Chap. 2, Lemma 3.3.2], and ∗ ∗ j i∗i (F) = 0 by [Chap. 2, Lemma 3.3.7]. ∗ Applying i , we see that η becomes an isomorphism since i∗ is fully faithful, and ψ becomes an isomorphism by [Chap. 2, Lemma 3.3.7]. Hence the conclusion follows.  100 2. THE FORMALISM OF SIX OPERATIONS

3.3.12. The localization property already implies left-adjointability along closed immersions and (smooth, closed)-base change, which is a big step towards (smooth, proper)-biadjointability: Lemma 3.3.13. Let D∗ be a premotivic category of coefficients satisfying S1-stability and localization. Then D∗ is (smooth, closed)-biadjointable.

Proof. For weak left-adjointability (Adjclosed), let i :Z ,→ S be a closed immersion and define a functor i! : D(S) → D(Z) by the formula ! ∗ ∗ (3.4) i (F) := i (Fib(F → j∗j (F))) (F ∈ D(S)).

∗ ∗ Write K = Fib(F → j∗j (F)). By [Chap. 2, Lemma 3.3.2] we have j (K) = 0. From ! ∗ the localization triangle it follows that i∗i (F) = i∗i (K) = K. We define a unit 2-morphism ! ! η : i∗i → id by the canonical morphism i∗i (F) = K → F. ! ∗ ∗ By [Chap. 2, Lemma 3.3.7] we have i i∗(F) = i (Fib(i∗F → j∗j i∗F)) = F. We define the ! counit ε : F → i i∗(F) = F to be the identity. It is straightforward to verify that η and ε verify the triangle identities defining an adjunction ! (i∗, i ). smooth For the properties (BCclosed), (Projclosed), and (BCclosed ), the proof is exactly the same as in the case of motivic spectra; see Corollaries 7.5.6, 7.6.4, 7.7.2. 

In particular, by right transposition from the localization triangle (3.3) ∗ ∗ j]j (F) → F → i∗i (F)(F ∈ D(S)), we obtain another exact triangle ! ∗ (3.5) i∗i (F) → F → j∗j (F)(F ∈ D(S)).

3.3.14. Another indication of the strength of the localization property is the observation that it forces Nisnevich separation. Lemma 3.3.15. Let D∗ be a premotivic category of coefficients satisfying S1-stability and localization. Then D∗ satisfies the property of Nisnevich separation (and hence a fortiori Zariski separation).

Proof. It suffices to show that for every Nisnevich square Q

V ×S U V p j U S, with p ´etaleof finite presentation and j a quasi-compact open immersion, the pair of functors (j∗, p∗) is conservative. By definition of Nisnevich square, there exists a closed immersion i :Z ,→ S complementary to j such that in the cartesian square V V × Z k S p q S Z i the morphism q is invertible. By the localization property, the pair (j∗, i∗) is conservative. This is equivalent to the ∗ ∗ ∗ ∗ ∗ ∗ conservativity of the pair (j , q i ) = (j , k p ), so the conclusion follows.  3. MOTIVIC CATEGORIES OF COEFFICIENTS 101

3.3.16. Recall that a nil-immersion is a closed immersion that induces an isomorphism on underlying reduced classical schemes. Definition 3.3.17. A category of coefficients D∗ satisfies topological invariance if for any ∗ nil-immersion i, the adjunction (i , i∗) is an equivalence.

In particular, this property means that the category of coefficients D∗ does not distinguish between a scheme S and its underlying classical scheme Scl (or even its underlying reduced classical scheme Scl,red). Topological invariance follows automatically from the stronger property of localization: Lemma 3.3.18. Let D∗ be a premotivic category of coefficients which is pointed and satisfies the localization property. Then D∗ satisfies topological invariance.

Proof. This follows immediately from the localization property applied to the closed immersion i, whose open complement is empty. 

3.4. Thom stability.

3.4.1. Let D∗ be a premotivic category of coefficients. Let S be a scheme, p :X → S a smooth morphism of finite presentation, and s :S ,→ X a section. Assume that p is separated, so that s is a closed immersion. The Thom suspension endofunctor associated to the pair (p, s) is defined as

thomp,s := p]s∗ : D(S) → D(S).

If D∗ is weakly right-adjointable with respect to closed immersions, then it admits a right adjoint, the Thom desuspension endofunctor thomp,s := s!p∗ : D(S) → D(S).

X/S When there is no risk of confusion, we will also write thomX/S := thomp,s and thom := thomp,s (when the latter is defined).

3.4.2. Let (p, s) be a pair as above and assume that p is ´etale.Then the closed immersion s is also an open immersion, i.e. it is an inclusion of a connected component. Assuming D∗ is additive, we may apply [Chap. 2, Lemma 3.3.4] to deduce that there is a canonical 2-isomorphism id = p]s] → thomp,s. That is: Lemma 3.4.3. Let D∗ be an additive category of coefficients. Let p be a separated ´etale morphism of finite presentation with a section s. Then the Thom suspension functor thomp,s coincides with the identity endofunctor of D(S).

3.4.4. Let p :X → S a smooth separated morphism of finite presentation, and s :S ,→ X a section. Given a morphism of schemes f :T → S, consider the base change square g Y X q p f T S. Let t :T ,→ Y be the induced section of q :Y → T. Then we have: 102 2. THE FORMALISM OF SIX OPERATIONS

Lemma 3.4.5. Suppose that D∗ is right-adjointable along closed immersions. Then there are canonical isomorphisms ∗ ∗ thomq,t ◦ f = f ◦ thomp,s, p,s q,t thom ◦ f∗ = f∗ ◦ thom .

Proof. The second isomorphism is the right transpose of the first. The first follows from left base change along smooth morphisms and right base change along closed immersions: we have ∗ ∗ ∗ f p]s∗ = q]g s∗ = q]t∗f . 

3.4.6. Suppose that there is a commutative diagram of schemes S s0 t0 (3.6) S0 t Y p0 q0 q p S s X S where p (resp. p0, q0) is a smooth separated morphism with section s (resp. s0, t0).

If the square is cartesian, the exchange 2-morphism q]t∗ → p]s∗ ((2.8)) gives rise to a 0 0 0 canonical 2-morphism p]q]t∗t∗ → p]s∗q]t∗ by horizontal composition. In other words, we have a canonical 2-morphism

(3.7) thomp0,s0 → thomp,s ◦ thomq0,t0 .

By construction, we have: Lemma 3.4.7. If D∗ satisfies bidirectional base change along (smooth, closed), then the 2- morphism (3.7) is invertible.

For example, this is true in the presence of localization ([Chap. 2, Lemma 3.3.13]). By [Chap. 2, Lemma 3.4.3] we further deduce: Corollary 3.4.8. Assume that D∗ satisfies the localization property. Suppose we have a commutative diagram as in (3.6). If q is ´etale,then there is a canonical 2-isomorphism

thomp0,s0 → thomp,s.

0 Proof. By 3.4.7 it suffices to show that the endofunctor thomq0,t0 is the identity. Since q is ´etale this follows from [Chap. 2, Lemma 3.4.3], which applies because Lemmas 3.3.15 and 3.3.5 ensure ∗ that D is additive. 

3.4.9. Suppose now that we have a commutative diagram S

(3.8) E0 t E p S E00 S where E, E0 and E00 are vector bundles over a scheme S, and the square is cartesian. (The unlabelled arrows ,→ are the zero sections of the respective vector bundles.) 3. MOTIVIC CATEGORIES OF COEFFICIENTS 103

In this case the 2-morphism (3.7) takes the form

(3.9) thomE/S → thomE00/S ◦ thomE0/S. By 3.4.7 this is invertible when D∗ satisfies bidirectional base change along the pair (smooth, closed).

3.4.10. We introduce the following property: Definition 3.4.11. The premotivic category of coefficients D∗ is Thom stable if the following condition holds: (Thom) For any pair (p, s) with p :X → S a smooth separated morphism and s :S → X a section, the Thom transformation thomX/S is an equivalence.

In practice, it suffices to consider the Thom transformation associated to the affine line: Lemma 3.4.12. Let D∗ be a premotivic category of coefficients satisfying the localization property. Then the following conditions are equivalent: (i) D∗ is Thom stable. (ii) For every scheme S and every vector bundle p :E → S, the Thom transformation thomE/S is an equivalence.

(iii) For every scheme S, the Thom transformation thom 1 is an equivalence. AS/S

Proof. Recall that the localization property implies (smooth, closed)-biadjointability ([Chap. 2, Lemma 3.3.13]) and Zariski separation ([Chap. 2, Lemma 3.3.15]). Let p :X → S be a separated smooth morphism of finite presentation with a section s. We begin by noting that the property of thomp,s being an equivalence is Zariski-local on S. Indeed, by bidirectional base change along the pair (smooth, closed) we obtain canonical isomorphisms ∗ ∼ j ◦ thomp,s −→ thomp0,s0 , 0 0 thomp ,s ◦ j∗ −→∼ j∗ ◦ thomp,s 0 0 0 where p :X ×S U → U is the base change of p and s is the induced section of p . Hence we conclude by Zariski separation. Now we show that (iii) implies (ii). By the above observation, claim (ii) reduces to the statement that the functor thom n is an equivalence for each n 0. But by (3.9) this functor AS /S > ◦n is identified with the n-fold composite (thom 1 ) , so we conclude by assumption. AS/S Next we show that (ii) implies (i). By [Chap. 1, Lemma 9.3.4], there exists a morphism ∗ q :X → NS/X to the conormal bundle of s, which is ´etaleon some open neighbourhood U ,→ X ∗ of the image of s, such that the section s is the base change of the zero section of Ns.

Let S0 := S ×X U denote the base change of S to U, and X0 := X ×S S0 the base change of X to S0. Using [Chap. 2, Corollary 3.4.8] we obtain (arguing as in [CD09, Prop. 2.4.11]), a 2-isomorphism

∗ thomX0/S0 = thomN /S0 S0/X0 and we conclude by (ii). 

3.4.13. Let D∗ be a premotivic category of coefficients. Let p :X → S be a smooth separated morphism of finite presentation, and s :S ,→ X a section. Definition 3.4.14. We define the Thom object associated to the pair (p, s) as the object

Th(p, s) := thomp,s(1S) := p]s∗(1S) in D(S). 104 2. THE FORMALISM OF SIX OPERATIONS

If p :E → S is a vector bundle with zero section s, we write ThS(E) := Th(p, s) for the associated Thom object.

3.4.15. Using the projection formulas, one observes: Lemma 3.4.16. Suppose that D∗ satisfies the right projection formula along closed immersions (Projclosed). Let S be a scheme, p :X → S a smooth separated morphism, and s :S ,→ X a section. Then we have canonical isomorphisms

thomp,s(F) = Th(p, s) ⊗S F, p,s thom (F) = HomS(Th(p, s), F) for each object F ∈ D(S).

Proof. The second isomorphism is the right transpose of the first. For the first, we use the left projection formula along smooth morphisms and the right projection formula along closed immersions to write ∗ ∗ ∗ ∗ ∗ p]s∗(1S) ⊗ F = p](s∗(1S) ⊗ p (F)) = p](s∗(1S ⊗S s p (F)) = p]s∗s p (F) = p]s∗(F), as desired (since p ◦ s = id). 

3.4.17. Assume that we have a commutative diagram as in (3.8). In this case, the canonical 2-morphism (3.9) takes the form 00 0 (3.10) ThS(E) → ThS(E ) ⊗S ThS(E ). Recall that this is invertible as soon as D∗ satisfies bidirectional base change with respect to the pair (smooth, closed).

3.4.18. Putting the above together, we see that in practice, in order to force Thom stability it 1 suffices to invert the Thom sheaves ThS(AS). Corollary 3.4.19. Let D∗ be a premotivic category of coefficients satisfying the localization property. Then the following conditions are equivalent: (i) D∗ is Thom stable. (ii) For every scheme S, every smooth separated morphism p :X → S, and every section s :S ,→ X, the Thom object Th(p, s) ∈ D(S) is invertible with respect to the monoidal product ⊗S. (iii) For every scheme S and every vector bundle p :E → S with zero section s :S ,→ E, the Thom object ThS(E) ∈ D(S) is invertible with respect to the monoidal product ⊗S. 1 (iv) For every scheme S, the Thom object ThS(AS) ∈ D(S) is invertible with respect to the monoidal product ⊗S.

Proof. By Lemmas 3.3.13 and 3.3.15, D∗ satisfies the right projection formula along closed immersions, and Zariski separation. Hence we may apply Lemmas 3.4.12 and 3.4.16 to conclude. 

We deduce that, in the presence of localization, Thom stability is stronger than S1-stability. Corollary 3.4.20. Let D∗ be a premotivic category of coefficients satisfying localization, Thom stability, and homotopy invariance. Then D∗ is S1-stable.

Proof. The proof is exactly as in the case of motivic spectra [Chap. 1, Lemma 8.3.3]. 

3.5. Motivic categories of coefficients. 3. MOTIVIC CATEGORIES OF COEFFICIENTS 105

3.5.1. The following is the natural analogue of the notion of motivic triangulated category of [CD09]: Definition 3.5.2. A premotivic category of coefficients D∗ is motivic if it satisfies the properties of homotopy invariance (Htp), Thom stability (Thom), and localization (Loc).

3.5.3. Assume that all schemes in the category Sch are quasi-compact and quasi-separated. In [Chap. 2, Paragraph 3.7] we will prove: Theorem 3.5.4. Let D∗ be a compactly generated category of coefficients. If D∗ is motivic, then it is (open, proper)-biadjointable.

3.6. Comparison with the axiomatic of Cisinski–Deglise.

3.6.1. Recall from [Lur16, Thm. 1.1.2.15] (cf. [GR16, Book-I.1, 5.1.2]) that for a stable (∞, 1)-category C, the underlying (1, 1)-category (C)ordn admits a triangulated structure. Let Arenastab denote the full subcategory of Arena spanned by stable arenas. Let Cattri V denote the (very large) (2, 1)-category of large triangulated categories, triangulated functors, and invertible triangulated natural transformations. We get: Lemma 3.6.2. The canonical functor of (∞, 1)-categories Arenastab → (1, 1)-Cat V, given on objects by the assignment C 7→ (C)ordn, lifts to a functor (3.11) Arenastab → Cattri V along the forgetful functor Cattri V → (1, 1)-Cat V.

Proof. Since the target of the desired functor is a (2, 1)-category, it is equivalent by adjunction to define a functor of (2, 1)-categories from the underlying (2, 1)-category of Arenastab, i.e. to define the functor on objects, 1-morphisms, and (invertible) 2-morphisms. For this, it suffices to note that an exact functor of stable (∞, 1)-categories induces a triangulated functor on triangulated categories, which follows directly upon inspection of the ordn definition of the triangulated structure on (C) . 

Further, when C is symmetric monoidal, the underlying (1, 1)-category (C)ordn is triangulated monoidal. That is, it admits a symmetric monoidal structure which is compatible with the triangulated structure, in the sense that the monoidal product is exact and commutes with arbitrary coproducts in each argument. This follows from [Lur16, Rem. 2.1.2.20] and the fact that the monoidal product on C commutes with arbitrary small colimits in each argument. Let Arenamonstab denote the full subcategory of Arenamon spanned by stable symmetric monoidal arenas. Let CattrimonV denote the (very large) (2, 1)-category of large triangulated monoidal categories, symmetric monoidal triangulated functors, and invertible symmetric monoidal triangulated natural transformations. We have: Lemma 3.6.3. The functor (3.11) lifts to a functor of (∞, 1)-categories (3.12) Arenamonstab → CattrimonV along the forgetful functor CattrimonV → Cattri V.

3.6.4. Given a premotivic category of coefficients D∗ :(Sch)op → Arenastab, we can restrict to the (1, 1)-category of classical schemes and obtain a functor (D∗)cl :(Schcl)op → Arenastab, 106 2. THE FORMALISM OF SIX OPERATIONS which clearly defines a premotivic category of coefficients (D∗)cl on Schcl. It is clear that, if D∗ is motivic, then so is its restriction (D∗)cl.

3.6.5. If D∗ is S1-stable, then by post-composing (D∗)cl with the functor (3.11), we obtain a functor of (2, 1)-categories (D∗)cl,tri :(Schcl)op → Cattri V Since the functor D∗ :(Sch)op → Arenastab is symmetric monoidal, (D∗)cl,tri factors through CattrimonV by [Chap. 2, Lemma 3.6.3]. It is immediate from the definitions that the functor (D∗)cl,tri defines a premotivic triangu- lated category in the sense of [CD09], which is motivic whenever (D∗)cl is. That is: Lemma 3.6.6. Let D∗ be an S1-stable premotivic category of coefficients D∗. Then the following hold: (i) The induced functor (D∗)cl,tri defines a premotivic triangulated category. (ii) If D∗ is a motivic category of coefficients, then the premotivic triangulated category (D∗)cl satisfies the localization, homotopy and stability properties (in the sense of loc. cit).

3.6.7. Conversely, the following lemma allows us to deduce properties of the motivic category of coefficients (D∗)cl from the corresponding properties of the motivic triangulated category (D∗)cl,tri. Lemma 3.6.8. Let D∗ be a motivic category of coefficients. Then the motivic category of coefficients (D∗)cl has weak right-adjointability (resp. right base change, the right projection formula, bidirectional base change) if and only if the premotivic triangulated category (D∗)cl,tri has the adjoint property (resp. the proper base change formula, the proper projection formula, the support property) in the sense of [CD09].

Proof. The properties in question involve statements of the following two forms: (1) A certain exact functor of stable (∞, 1)-categories is an equivalence. (2) A certain exact functor of stable arenas commutes with small colimits (i.e. admits a right adjoint). (3) A certain 2-morphism between functors of (∞, 1)-categories is invertible. For statements of the first form it is sufficient to recall that an exact functor of stable (∞, 1)- categories is an equivalence if and only if it induces a triangulated equivalence of underlying (1, 1)-categories. For statements of the second form, let u be an exact functor of stable (∞, 1)-categories. For u to commute with small colimits, it suffices that it commutes with arbitrary direct sums, a property which can be checked on underlying (1, 1)-categories. Statements of the third form can be checked object-wise in the underlying (1, 1)-categories. 

3.6.9. Write opencl (resp. propercl) for the intersection of the class open (resp. proper) with the subcategory Schcl ⊂ Sch of classical schemes. Recall the following theorem of Ayoub [Ayo07], which is the classical version of [Chap. 2, Theorem 3.5.4]: Theorem 3.6.10. Let (D∗)cl be a motivic category of coefficients which is compactly generated2. ∗ cl Then (D ) is (opencl, propercl)-biadjointable.

2I.e., the category D(S) is compactly generated for each classical scheme S. 3. MOTIVIC CATEGORIES OF COEFFICIENTS 107

Proof. For noetherian (classical) bases, the claim follows from [CD09, Thms. 2.4.26 and 2.4.28], using [Chap. 2, Lemma 3.6.8] to translate. To generalize to the case of quasi-compact quasi-separated (classical) bases, one may use the argument in the proof of [Hoy15, Prop. C.13], which clearly works mutatis mutandis in the setting of any compactly generated motivic category of coefficients. 

3.7. Biadjointability of motivic categories of coefficients.

3.7.1. In this paragraph we will prove [Chap. 2, Theorem 3.5.4], which states that any (compactly generated) motivic category of coefficients is (open, proper)-biadjointable. Our starting point is the result of Ayoub, [Chap. 2, Theorem 3.6.10], which says that (D∗)cl is (opencl, propercl)-biadjointable. By the localization property, we also have that D∗ satisfies topological invariance ([Chap. 2, Lemma 3.3.18]) and (smooth, closed)-biadjointability ([Chap. 2, Lemma 3.3.13]). The following sequence of lemmas will then demonstrate (open, proper)-biadjointability.

3.7.2. First, we check that (Adj ) is equivalent to (Adj ): proper propercl Lemma 3.7.3. Let D∗ be a premotivic category of coefficients satisfying S1-stability, topological invariance, and weak right-adjointability along closed. Then D∗ is weakly right-adjointable along ∗ cl a morphism f if and only if (D ) is weakly right-adjointable along fcl.

Proof. Consider the commutative square

k Tcl T

fcl f i Scl S.

∗ By topological invariance, the adjunction (k , k∗) is an equivalence. By weak closed-adjointability, ! ! i∗ (resp. k∗) admits a right adjoint i (resp. k ). ! ! ! ! If (fcl) is a right adjoint to (fcl)∗, then it is easy to verify that f := k∗(fcl) i is a right adjoint to f∗. ! ! ! ! Conversely if f is a right adjoint to f∗, then (fcl) := k f i∗ is a right adjoint to (fcl)∗. 

The next lemma says that (BCproper) can be checked on underlying classical schemes: Lemma 3.7.4. Let D∗ be an S1-stable premotivic category of coefficients satisfying topological invariance and right base change along closed. Then D∗ satisfies right base change along a ∗ cl morphism g if and only if (D ) satisfies right base change along gcl.

Proof. Suppose there is a cartesian square of schemes

f 0 T0 S0

(3.13) g0 g f T S 108 2. THE FORMALISM OF SIX OPERATIONS with g and g0 proper. Consider the induced cartesian square

0 0 fcl 0 Tcl Scl (3.14) 0 g gcl cl

fcl Tcl Scl in the category of classical schemes. We will show that the exchange 2-morphism ∗ 0 0 ∗ α : f g∗ → (g )∗(f ) is invertible if and only if the exchange 2-morphism ∗ 0 0 ∗ β :(fcl) (gcl)∗ → (gcl)∗(fcl) is invertible. Consider the commutative cube 0 0 k 0 Tcl T g0 0 gcl 0 k fcl Tcl T f 0

0 0 i 0 f Scl S fcl g gcl i Scl S. ∗ 0∗ 0 ∗ 0∗ 0 By topological invariance, the adjunctions (i , i∗), (i , i∗), (k , k∗), (k , k∗) are all equivalences. We claim that there is a canonical isomorphism of natural transformations ∗ 0 β = k ∗ α ∗ (i )∗, where ∗ denotes horizontal composition, from which the claim follows. ∗ ∗ 0 ∗ ∗ ∗ ∗ Indeed, note that there is a canonical identification k f g∗(i )∗ = k f i∗(gcl)∗ = (fcl) i i∗(gcl)∗ = ∗ (fcl) (gcl)∗. ∗ 0 0 ∗ 0 ∗ 0 0 0 ∗ On the other hand, there is a canonical identification k (g )∗(f ) (i )∗ = k (g )∗(k )∗(fcl) = ∗ 0 0 ∗ 0 0 ∗ 0 k k∗(gcl)∗(fcl) = (gcl)∗(fcl) by the closed base change formula for i . The fact that the induced natural transformation ∗ 0 ∗ 0 0 ∗ k ∗ α ∗ (i )∗ :(fcl) (gcl)∗ → (gcl)∗(fcl) coincides with α follows by careful inspection from the constructions of the exchange transfor- mations α and β, respectively. 

The following lemma says that (Projproper) can be checked on underlying classical schemes: Lemma 3.7.5. Let D∗ be an S1-stable premotivic category of coefficients satisfying topological invariance and the right projection formula along closed. Then D∗ satisfies the right projection ∗ cl formula along a morphism f if and only if (D ) satisfies the right projection formula along fcl.

Proof. We will show that the canonical morphism ∗ α(F, G): f∗(F ⊗T f (G)) → f∗(F) ⊗S G is invertible for all objects F ∈ D(T) and G ∈ D(S), if and only if the canonical morphism ∗ β(F0, G0):(fcl)∗(F0 ⊗T (fcl) (G0)) → (fcl)∗(F0) ⊗S G0 3. MOTIVIC CATEGORIES OF COEFFICIENTS 109 is invertible for all objects F0 ∈ D(Tcl) and G0 ∈ D(Scl). Consider the commutative square

i0 Tcl T

fcl f i Scl S.

∗ ∗ By topological invariance the adjunctions (i , i∗) and (k , k∗) are equivalences. ∗ In particular, given objects F0 ∈ D(Tcl) and G0 ∈ D(Scl), we can write F0 = k (F) and ∗ G0 = i (G), where F := k∗(F0) and G := i∗(G0), respectively. The claim follows from the fact that, for each F0 and G0, the morphism α(F, G) is canonically identified with i∗(β(F0, G0)). ∗ ∗ Indeed, note that we have i∗((fcl)∗(F0 ⊗(fcl) (G0))) = f∗(F⊗f G), and i∗((fcl)∗(F0)⊗G0) = ∗ ∗ i∗((fcl)∗k (F) ⊗ i (G)) = f∗(F) ⊗ G by the right projection formula for i. 

open 3.7.6. Finally, the following lemma says that (BCproper) can be checked on underlying classical schemes:

Lemma 3.7.7. Let D∗ be an S1-stable premotivic category of coefficients satisfying topological invariance and bidirectional base change along the pair (open, closed). Then D∗ satisfies bidirec- tional base change along (open, proper) if and only if (D∗)cl satisfies bidirectional base change along (opencl, propercl).

Proof. Suppose we have a cartesian square of schemes

f 0 V U

(3.15) j0 j f T S with f and f 0 proper, and j and j0 open immersions, and let

0 fcl Vcl Ucl (3.16) 0 jcl jcl

fcl Tcl Scl denote the induced cartesian square in the category of classical schemes. We will show that the exchange 2-morphism

0 0 α : j](f )∗ → f∗(j )] is invertible if and only if the exchange 2-morphism

0 0 β :(jcl)](fcl)∗ → (fcl)∗(jcl)] is invertible. 110 2. THE FORMALISM OF SIX OPERATIONS

Consider the commutative cube 0 jcl Vcl Tcl k k0 fcl 0 fcl V T f 0 f 0 jcl Ucl Scl f i 0 i j U S.

0 ∗ 0 Using (open, closed)-base change and the fact that the adjunction ((k ) , (k )∗) is an equiva- lence by topological invariance, one sees that there is a canonical isomorphism of 2-morphisms 0 ∗ α = i∗ ∗ β ∗ (k ) , from which the claim follows.  4. THE FORMALISM OF SIX OPERATIONS 111

4. The formalism of six operations

4.1. Schematic correspondences.

4.1.1. Let C be an (∞, 1)-category, and let c and c0 be two objects in C. A correspondence from c to c0 is a diagram

f d c g c0 in C. We will also write such a datum as a triple (d, f, g). Let (d, f, g) and (d0, f 0, g0) be two correspondences from c to c0.A morphism of correspon- dences from (d, f, g) to (d0, f 0, g0) is a morphism h : d → d0 in C together with a commutative diagram d f h

f 0 g d0 c

g0 c0 in C. From [GR16, Book-V.1], we have:

Proposition 4.1.2. There is an (∞, 2)-category whose objects are objects of C, 1-morphisms are correspondences, and 2-morphisms are morphisms of correspondences.

4.1.3. Let Arrsch = Arrows(Sch) denote the (∞, 1)-category of morphisms in Sch. To avoid ambiguity, we will use the term schematic arrow when we want to view a morphism of schemes as an object of Arrsch.

Definition 4.1.4. A schematic correspondence is a correspondence in the category of schematic arrows.

We will write Corrsch for the (∞, 2)-category of schematic correspondences, and Corrsch for its underlying (∞, 1)-category.

4.1.5. Let horiz and vert be classes of morphisms in Arrsch. Given schematic arrows f and f 0, a schematic correspondence (g, α, β) from f to f 0 is of type (horiz, vert) if the morphism α (resp. β) is contained in horiz (resp. vert).

Let Corrschhoriz;vert (resp. Corrschhoriz;vert ) denote the sub-(∞, 2)-category of Corrsch (resp. sub-(∞, 1)-category of Corrsch) where the 1-morphisms are spanned by correspondences of type (horiz, vert). Let diag be another class of morphisms in Arrsch. We will write Corrschdiag (resp. diag Corrschhoriz;vert ) for the sub-(∞, 2)-category of Corrsch (resp. of Corrschhoriz;vert ) where the 2-morphisms are spanned by morphisms of correspondences (g, α, β) → (g0, α0, β0) such that the underlying morphism of schematic arrows γ : g → g0 is contained in diag. 112 2. THE FORMALISM OF SIX OPERATIONS

4.1.6. Given a property of morphisms of schemes (P), we will say that a morphism of schematic arrows α : f → g has the property (P) if it is of the form

f T S f S S where the morphism f has the property (P). For example, we may speak of morphisms of schematic arrows being open immersions, proper, separated, or of finite type. Let open (resp. proper, sep, all) denote3 the class of open immersions (resp. proper morphisms, separated morphisms of finite type, all morphisms) of schematic arrows. proper In the sequel, a special rˆole will be played by the (∞, 2)-category Corrschsep:all , where all denotes the class of all morphisms of schematic arrows.

4.2. The extension theorem.

4.2.1. Recall from [Chap. 0, Paragraph 3.4] the (∞, 2)-category Arenamod of pairs (O, C), with O a symmetric monoidal arena and C an O-module arena. The following important theorem will be a straightforward application of the technology developed in [GR16]: Theorem 4.2.2. Suppose that every scheme S in Sch is quasi-compact and quasi-separated. Let D∗ be an (open, proper)-biadjointable category of coefficients defined on Sch. Then there exists a unique extension of D∗ to a symmetric monoidal functor of (∞, 2)-categories

∗ proper 2-op (4.1) D! : Corrschsep;all → (Arenamod) .

Combining this with [Chap. 2, Theorem 3.5.4], we obtain immediately: Corollary 4.2.3. Suppose that every scheme S in Sch is quasi-compact and quasi-separated. Let D∗ be a motivic category of coefficients defined on Sch. Then there exists a unique extension of D∗ to a symmetric monoidal functor of (∞, 2)-categories

∗ proper 2-op (4.2) D! : Corrschsep;all → (Arenamod) .

The rest of this section will be devoted to the proof of [Chap. 2, Theorem 4.2.2].

4.2.4. Before discussing the extension to schematic correspondences, we start by noting that we can extend any category of coefficients D∗ to a functor of (∞, 1)-categories ∗ op (4.3) DArrsch :(Arrsch) → Arenamod defined as the composite (Arrsch)op = Arrows((Sch)op) → Arrows(Arenamon) → Arenamod. The first functor is obtained from D∗ :(Sch)op → Arenamon by applying Arrows(−). The second is the canonical functor of (3.10), which sends a symmetric monoidal morphism of arenas u : O → O0 to the pair (O, O0), where O0 is viewed as an O-module via u.

3This is a slight abuse of notation, but it will always be clear from the context whether we are referring to a class of morphisms in Sch or Arrsch. 4. THE FORMALISM OF SIX OPERATIONS 113

∗ 4.2.5. The functor DArrsch can be described informally as follows: On objects, it sends a schematic arrow f :T → S to the pair (D(S), D(T)), where D(T) is viewed as a D(S)-module via the symmetric monoidal functor f ∗. On morphisms, it sends a morphism of schematic arrows α : g → f, given by a commutative square f 0 T0 S0

g0 g f T S, to the morphism α∗ :(D(S), D(T)) → (D(S0), D(T0)) defined by the pair (g∗, (f 0)∗).

4.3. Biadjointability on schematic arrows. In this paragraph, we show that the extension of D∗ to schematic arrows, i.e. the functor ∗ op DArrsch :(Arrsch) → Arenamod, still satisfies the open-base change and proper-base change properties. Another way to say this is as follows. One can consider the evident analogues of the properties weakly left-adjointable, left base change, weakly right-adjointable, right base change, and bidirectional base change, for functors valued in an arbitrary 2-category. It turns out that ∗ ∗ ∗ the operation D 7→ DArrsch preserves them, for any biadjointable D .

4.3.1. The following verifies the properties of weak left-adjointability and left base change for ∗ DArrsch :

Lemma 4.3.2. (i) For every open immersion β : j → idS of schematic arrows given by a commutative square j T S j S S, the induced morphism ∗ ∗ β := DArrsch (β):(D(S), D(S)) → (D(S), D(T)) admits a left adjoint β] in the 2-category Arenamod. (ii) For every cartesian square of schematic arrows

0 j0 α j

(4.4) β0 β α idS0 idS with β and β0 open immersions, the induced commutative square in Arenamod

∗ ∗ α ∗ 0 DArrsch (f) DArrsch (f )

β∗ (β0)∗ 0 ∗ ∗ (α ) ∗ 0 DArrsch (j) DArrsch (j ), ∗ obtained by applying the functor DArrsch , is vertically left-adjointable.

Dually, we have: 114 2. THE FORMALISM OF SIX OPERATIONS

Lemma 4.3.3. (i) For every proper morphism α : f → idS of schematic arrows, given by a commutative square f T S

f S S, the induced morphism α∗ :(D(S), D(S)) → (D(S), D(T)) admits a right adjoint α∗ in the 2-category Arenamod. (ii) For every cartesian square of schematic arrows

0 g0 α g

β0 β f 0 α f, with α and α0 proper, the induced commutative square in Arenamod

∗ ∗ α ∗ 0 DArrsch (f) DArrsch (f )

β∗ (β0)∗ 0 ∗ ∗ (α ) ∗ 0 DArrsch (g) DArrsch (g ), ∗ obtained by applying the functor DArrsch , is horizontally right-adjointable.

Finally, we have the bidirectional base change property: Lemma 4.3.4. For all cartesian squares of schematic arrows

0 g0 α f 0

β0 β g α f with β and β0 open immersions, and α and α0 proper, the induced commutative square in Arenamod (q0)∗ (D∗, D∗)(f 0) (D∗, D∗)(g0)

p 0 ] p] q∗ (D∗, D∗)(f) (D∗, D∗)(g0), ∗ obtained by applying the functor DArrsch and vertically passing to left adjoints, is horizontally right-adjointable.

4.3.5. All three of these lemmas are purely formal exercises. We will only prove the first statement; the second is completely dual, and the third is also similar. Let β : j → f be an open immersion given by a commutative square as above. Recall that the morphism β∗ :(D(S), D(S)) → (D(S), D(T)) ∗ is defined by the pair (idD(S), j ). We define a morphism in Arenamod

β] :(D(S), D(T)) → (D(S), D(S)) given by the triple (idD(S), j]). 4. THE FORMALISM OF SIX OPERATIONS 115

∗ We will show that β] is left adjoint to β , by constructing unit and counit morphisms satisfying the triangle identities.

∗ 4.3.6. Let us define the unit 2-morphism η : id(D(S),D(T)) → β β] in Arenamod. This is the 2-morphism ∗ η : (idD(S), idD(T)) → (idD(S), j j]) by the data:

• The natural transformation idD(S) → idD(S) given by the identity. ∗ • The D(S)-linear natural transformation idD(T) → j j] given by the unit of the adjunc- ∗ tion (j], j ).

∗ 4.3.7. Let us define the counit 2-morphism ε : β]β → id(D(S),D(S)) in Arenamod. This is the 2-morphism in Arenamod ∗ ε : (idD(S), j]j ) → (idD(S), idD(S)) by the data:

• The natural transformation idD(S) → idD(S) given by the identity. ∗ • The D(S)-linear natural transformation j]j → idD(S) given by the counit of the ∗ adjunction (j], j ).

∗ 4.3.8. It is immediate that the data of the 2-morphisms η and ε defines an adjunction (β], β ) in the 2-category Arenamod.

4.3.9. We next verify the base change property, i.e. statement (ii) of [Chap. 2, Lemma 4.3.2]. 0 0 Let β : j → idD(S) and β : j → idD(S0) be given by commutative squares

j j0 T S T0 S0

j j0 S S, S0 S0, respectively. 0 0 Let α : idD(S0) → idD(S) and α : j → j be given by commutative squares

j0 S0 S0 T0 S0 f f g f j S S, T S, respectively. Assuming that the square (4.4) is cartesian, we have to show that the exchange 2-morphism 0 0 ∗ ∗ (4.5) (β )](α ) → α β] is invertible. Note that (4.4) induces a cartesian square of schemes

j0 T0 S0 (4.6) g f j T S. 116 2. THE FORMALISM OF SIX OPERATIONS

∗ ∗ ∗ 4.3.10. The composite α β] is given by the pair (f , f j]). 0 0 ∗ ∗ 0 ∗ The composite (β )](α ) is given by the pair (f , (j )]g ). The 2-morphism (4.5) is given by the following data:

• The natural transformation f ∗ → f ∗ given by the identity. ∗ 0 ∗ • The D(S)-linear transformation f j] → (j )]g given by the exchange transformation associated to the cartesian square (4.6).

By the axiom (BCopen) it follows that this 2-morphism is invertible.

4.4. The extension theorem, proof. In this paragraph we will prove [Chap. 2, Theo- rem 4.2.2].

4.4.1. We begin with the functor ∗ op DArrsch :(Arrsch) → Arenamod defined in (4.3). ∗ The first part of the construction of the 2-functor D! (4.2) consists of the extension of ∗ DArrsch to a symmetric monoidal functor of (∞, 2)-categories ∗ iso (4.7) D] : Corrschopen;all → Arenamod. We will obtain this extension by using [GR16, Book-V.1, Thm. 3.2.2(b)], which applies because of the following two observations: (1) The triple (all, open, open) satisfies the conditions of [GR16, Book-V.1, 1.1.1]. That is to say, the class of open immersions of schematic arrows is stable under base change and satisfies the 2-of-3 property. This follows immediately from the respective properties for the class of open immersions of schemes. ∗ (2) The functor DArrsch satisfies the right Beck-Chevalley condition [GR16, Book-V.1, Def. 3.1.5] with respect to the class of open immersions. This is the content of [Chap. 2, Lemma 4.3.2].

∗ 4.4.2. Next, we want to extend D] (4.7) to a symmetric monoidal functor of (∞, 2)-categories ∗ proper 2-op (4.8) D! : Corrschsep;all → (Arenamod) .

∗ For this, we flip the directions of the 1- and 2-morphisms in D] to obtain a 2-functor iso 1&2-op (4.9) Corrschall;open → (Arenamod) .

In order to apply [GR16, Book V.1, Thm. 5.2.4], we need to check the following: (1) The classes proper and sep satisfy the conditions of [GR16, Book-V.1, 1.1.1 and 5.1.1]. That is to say, the class proper is stable under base change, and the class sep is stable under base change and satisfies the 2-of-3 property. (2) The classes open and proper satisfy the condition of [GR16, Book-V.1, 5.1.2]. That is to say, every morphism in the intersection of the classes open and proper is a monomorphism. This follows immediately from the corresponding fact for the classes open and proper in Sch. In fact, recall that every open immersion is a monomorphism. (3) The classes proper, open and all satisfy the condition of [GR16, Book-V.1, 5.1.3]. That is to say, for every morphism γ of schematic arrows which is separated and of finite type, the 4. THE FORMALISM OF SIX OPERATIONS 117 category of commutative diagrams γ f g, β α

f 0 with β an open immersion and α proper, is contractible. This follows immediately from the corresponding statement for morphisms of schemes, which is [Chap. 0, Proposition 6.3.4]. ∗ (4) The functor DArrsch satisfies the left Beck-Chevalley condition of [GR16, Book-V.1, Def. 3.1.2] with respect to the class of proper morphisms. This is the content of [Chap. 2, Lemma 4.3.3]. (5) The functor (4.9) satisfies the condition of [GR16, Book V.1, 5.2.2]. This is the content of [Chap. 2, Lemma 4.3.4]. Hence we may apply [GR16, Thm. 5.2.4] to obtain a 2-functor proper 1&2-op Corrschall;sep → (Arenamod) . By flipping the directions of 1-morphisms we obtain the 2-functor ∗ proper 2-op D! : Corrschsep;all → (Arenamod) which is the extension desired. 118 2. THE FORMALISM OF SIX OPERATIONS

5. Example: the stable motivic homotopy category

5.1. The functor SH∗. In this paragraph we will define the stable motivic homotopy category, introduced in Chapter 1, as a category of coefficients in the sense of [Chap. 2, Sect. 2]. Here the notation Sch will be used for the category of quasi-compact quasi-separated schemes.

proper 5.1.1. Let Corrschsep;all denote the (∞, 2)-category of schematic correspondences (see [Chap. 2, Paragraph 4.1]). In this section we will show:

Theorem 5.1.2. The assignment S 7→ SH(S) lifts to a symmetric monoidal functor of (∞, 2)- categories

∗ proper 2-op (5.1) SH! : Corrschsep;all → (Arenamod) .

5.1.3. We begin by lifting the assignment S 7→ SH(S) to a motivic category of coefficients. Consider the canonical functor

Arrows(Sch) → Sch sending a morphism of schemes f :T → S to its target S. By abstract nonsense (see [GR16, Chap. I.1]), this is a cartesian fibration which corresponds, via straightening/unstraightening, to the presheaf of categories (Sch)op → (∞, 1)-Cat given object-wise by the assignment S 7→ Sch/S.

5.1.4. Let Sm denote the full subcategory of Arrows(Sch) spanned by smooth morphisms of schemes. Since smoothness is stable under base change, it is easily verified that the composite

Sm ,→ Arrows(Sch) → Sch is also a cartesian fibration. This corresponds by straightening/unstraightening to a presheaf of categories (Sch)op → (∞, 1)-Cat given object-wise by the assignment

S 7→ Sm/S.

5.1.5. Applying the canonical functor C 7→ P(C) object-wise, we obtain a presheaf of arenas (Sch)op → Arena given on objects by the assignment

S 7→ Spc(S) and on morphisms by f 7→ f ∗. Note that this lifts to a presheaf of symmetric monoidal arenas: each category Spc(S) is cartesian monoidal, and the morphisms f ∗ are symmetric monoidal (i.e. commute with finite products).4 Hence we have a functor (Sch)op → Arenamon.

4Note that in the cartesian monoidal case, symmetric monoidality is a property of functors, not a structure. 5. EXAMPLE: THE STABLE MOTIVIC HOMOTOPY CATEGORY 119

5.1.6. Consider the category of pairs (C, S), with C an arena and S an essentially small set of morphisms; see [Chap. 0, Paragraph 2.6]. Since the classes of Nisnevich covers and A1-projections are stable under base change, respectively, one sees that the functor (Sch)op → Arenamon lifts to (commutative monoids in) this category of pairs, where we send a scheme S to the category Spc(S) together with the small set of Nisnevich-local and A1-local isomorphisms. Applying the symmetric monoidal functor given by object-wise localization of pairs (see loc. cit.), we obtain a functor (Sch)op → Arenamon given on object by the assignment S 7→ H(S).

op 5.1.7. Applying the functor C 7→ C• object-wise, we obtain a functor (Sch) → Arenamon given on objects by the assignment S 7→ H(S)•.

5.1.8. Fix a family of pointed fibred spaces (TS)S as in [Chap. 1, Paragraph 5.3]. The properties of the “formal inversion” H(S)• 7→ SHT(S) studied in [Rob14] provide a functor (Sch)op → Arenamod, or equivalently a symmetric monoidal functor (Sch)op → Arena, given on objects by the assignment S 7→ SHT(S). See especially §9.1 of loc. cit.. This is the category of coefficients desired.

∗ ∗ 5.2. The 2-functor SH! . In this paragraph we obtain the 2-functor SH! encoding the formalism of six operations on motivic spectra.

5.2.1. First, we have that the category of coefficients SH∗ is left-adjointable along smooth morphisms. This was demonstrated in [Chap. 1, Sect. 6]. Hence SH∗ is premotivic.

5.2.2. We have homotopy invariance for SH∗ by construction, using point (iii) of [Chap. 2, Lemma 3.2.4].

1 1 5.2.3. We now fix the family (TS)S = (PS)S, and write SH(S) = SHP1 (S). Recall that PS 1 denotes, by abuse of notation, the pointed motivic space (MS(PS), ∞). 1 1 Recall also that there is a canonical identification PS = ThS(AS) for each S (see [Chap. 1, Corollary 8.2.4]). Hence we have Thom stability for SH∗ by construction, using point (iv) of [Chap. 2, Corollary 3.4.19].

5.2.4. We have the localization property for SH∗; this was the main result of Chapter 1 (see [Chap. 1, Paragraph 7.4]).

5.2.5. In summary, the category of coefficients SH∗ is motivic. By [Chap. 2, Corollary 4.2.3], we obtain the symmetric monoidal functor of (∞, 2)-categories ∗ proper 2-op (5.2) SH! : Corrschsep;all → (Arenamod) . desired, a unique extension of SH∗, encoding the formalism of six operations on motivic spectra.

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